Comprehensive Lecture Notes on Linear Algebra and Trigonometry

Course Structure and Assessment Policy

Course Title: Linear Algebra and Trigonometry (GL/ES/GM – 153)
Academic Institution: University of Mines and Technology, Tarkwa, School of Railway and Infrastructural Development, Department of Mathematical Sciences
Assessment Framework:
- Continuous Assessment ( $40\%$
- Attendance: Random attendance checks constitute $10\%$ of the continuous assessment.
- Assignments and Presentations: Group assignments, individual assignments, and group presentations constitute $10\%$ of the continuous assessment.
- Quizzes and Class Tests: Announced and unannounced quizzes constitute $20\%$ of the continuous assessment.
- Note: The course instructor determines the exact frequency and schedule of all assignments and quizzes.
- End of Semester Examination ( $60\%$
- A comprehensive written examination evaluating all covered topics.
- Student Evaluation of Instructor: Administered at the end of the semester via a structured questionnaire to assess instructional quality and course delivery.

Complex Numbers: Foundations and Algebraic Structure

Historical Evolution of Number Systems:
- Natural numbers ( $\mathbb{N}$ ) were augmented with zero to form whole numbers, and negative numbers were introduced to form integers ( $\mathbb{Z}$ ).
- Linear equations of the form $qx = p$ necessitated rational numbers ( $\mathbb{Q}$ ).
- Incommensurable geometric lengths necessitated irrational numbers; rationals and irrationals together form real numbers ( $\mathbb{R}$ ).
- Real numbers cannot solve equations such as $x^2 + 1 = 0$ or $x^2 = -1$ . This necessitated extending the real system to complex numbers ( $\mathbb{C}$ ).
Fundamental Definition of the Imaginary Unit:
- The imaginary unit $i$ satisfies the identity: $i^2 = -1$ $i = \sqrt{-1}$
- Solutions to $x^2 + 1 = 0$ are given by $x = \pm \sqrt{-1} = \pm i$ .
- Square roots of negative quantities factor as $\sqrt{-k} = \sqrt{-1 \times k} = \sqrt{-1}\sqrt{k} = i\sqrt{k}$ .
Standard Algebraic Form of a Complex Number:
- Any number expressible as $z = a + bi$ , where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ , is a complex number.
- Real Part: Expressed as $\text{Re}(z) = a$ .
- Imaginary Part: Expressed as $\text{Im}(z) = b$ .
- Purely Imaginary Numbers: If $\text{Re}(z) = a = 0$ , then $z = bi$ is called purely imaginary (e.g., $3i$ , $-10i$ , $\pi i$ ).
- Real Numbers as a Subset: When $\text{Im}(z) = b = 0$ , $z = a \in \mathbb{R}$ , confirming $\mathbb{R} \subset \mathbb{C}$ .
Equality of Complex Numbers:
- Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal if and only if their corresponding real parts and imaginary parts are identical: $a + bi = c + di \iff a = c \quad \text{and} \quad b = d$
Quadratic Equations with Complex Roots:
- For $ax^2 + bx + c = 0$ , the quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Worked Example: Solve $x^2 - 6x + 13 = 0$ :
- Identify coefficients: $a = 1$ , $b = -6$ , $c = 13$ .
- Substitute into formula: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(13)}}{2(1)}$ $x = \frac{6 \pm \sqrt{36 - 52}}{2} = \frac{6 \pm \sqrt{-16}}{2}$ $x = \frac{6 \pm 4i}{2} = 3 \pm 2i$

Complex Conjugates and Geometric Representation

Complex Conjugate Definition:
- For any complex number $z = a + bi$ , its complex conjugate is denoted by $\bar{z}$ and defined as: $\bar{z} = a - bi$
- Conjugate pairs arise naturally as roots of real polynomials (e.g., $3 + 2i$ and $3 - 2i$ ).
Properties of the Complex Conjugate:
- A complex number is real if and only if it equals its conjugate: $z = \bar{z} \iff z \in \mathbb{R}$ .
- Verification: If $z = 5 = 5 + 0i$ , then $\bar{z} = 5 - 0i = 5$ .
- A complex number is purely imaginary if and only if its conjugate is its negative: $\bar{z} = -z$ .
- Verification: If $z = 7i = 0 + 7i$ , then $\bar{z} = 0 - 7i = -7i = -z$ .
- The involution property holds: the conjugate of a conjugate is the original number: $\overline{(\bar{z})} = z$
- Distribution over addition: $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$
- Distribution over multiplication: $\overline{z_1 \cdot z_2} = \bar{z}_1 \cdot \bar{z}_2$
- Distribution over division: $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}$
Geometrical Representation: The Complex Plane (Argand Diagram):
- A complex number $z = a + bi$ corresponds to an ordered pair of real numbers $(a, b)$ plotted in a Cartesian coordinate system.
- Horizontal Axis ( $x$ -axis): The Real Axis ( $\text{Re}$ -axis), representing $a$ .
- Vertical Axis ( $y$ -axis): The Imaginary Axis ( $\text{Im}$ -axis), representing $b$ .
- Points $z = a + bi$ and $\bar{z} = a - bi$ represent reflections across the real axis.

Modulus (Absolute Value) of a Complex Number:
- The modulus represents the Euclidean distance from the origin $O(0, 0)$ to the point $P(a, b)$ .
- By the Pythagorean theorem on right triangle $OMP$ with base $OM = a$ and perpendicular $MP = b$ : $|z| = \sqrt{OM^2 + MP^2} = \sqrt{a^2 + b^2}$
- Algebraic relation with conjugate: $z \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 = |z|^2$
Essential Modulus Inequalities and Properties:
- $|z_1 z_2| = |z_1| |z_2|$
- $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ , provided $z_2 \neq 0$
Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$
Reverse Triangle Inequality: $|z_1 - z_2| \ge |z_1| - |z_2|$

Complex Arithmetic

Addition of Complex Numbers:
- Add corresponding real and imaginary parts: $(a + bi) + (c + di) = (a + c) + (b + d)i$
- Example 1: If $z_1 = 6 + 9i$ and $z_2 = 12 + 7i$ : $z_1 + z_2 = (6 + 12) + (9 + 7)i = 18 + 16i$
- Example 2: If $z_1 = 4 + 7i$ and $z_2 = 1 - 6i$ : $z_1 + z_2 = (4 + 1) + (7 - 6)i = 5 + i$
Subtraction of Complex Numbers:
- Subtract corresponding real and imaginary parts: $(a + bi) - (c + di) = (a - c) + (b - d)i$
- Example 1: If $z_1 = 6 + 9i$ and $z_2 = 12 + 7i$ : $z_1 - z_2 = (6 - 12) + (9 - 7)i = -6 + 2i$
- Example 2: If $z_1 = 4 + 7i$ and $z_2 = 1 - 6i$ : $z_1 - z_2 = (4 - 1) + (7 - (-6))i = 3 + 13i$
Multiplication of Complex Numbers:
- Perform polynomial binomial expansion applying $i^2 = -1$ : $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$
- Example: Evaluate $(3 + 2i)(4 + 5i)$ : $(3 + 2i)(4 + 5i) = 3(4 + 5i) + 2i(4 + 5i) = 12 + 15i + 8i + 10i^2 = 12 + 23i - 10 = 2 + 23i$
Division of Complex Numbers:
- Multiply numerator and denominator by the complex conjugate of the denominator: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \left(\frac{bc - ad}{c^2 + d^2}\right)i$
- Example: Divide $3 + i$ by $4 - 2i$ : $\frac{3 + i}{4 - 2i} = \frac{(3 + i)(4 + 2i)}{(4 - 2i)(4 + 2i)} = \frac{12 + 6i + 4i + 2i^2}{16 - 4i^2} = \frac{12 + 10i - 2}{16 + 4} = \frac{10 + 10i}{20} = \frac{1}{2} + \frac{1}{2}i$
Extraction of Square Roots of Complex Numbers:
- To find $\sqrt{x + yi}$ , set $a + bi = \sqrt{x + yi}$ and square both sides: $(a + bi)^2 = a^2 - b^2 + 2abi = x + yi$
- Equate real and imaginary components: $a^2 - b^2 = x$ $2ab = y$
- Worked Example: Compute $\sqrt{35 - 12i}$ :
- Let $\sqrt{35 - 12i} = a + bi$ .
- Expanding: $a^2 - b^2 + 2abi = 35 - 12i$ .
- Equate components: $a^2 - b^2 = 35$ and $2ab = -12 \implies ab = -6$ .
- Solving the system yields $a = 6, b = -1$ or $a = -6, b = 1$ .
- Therefore, the square roots are $\pm(6 - i)$ .

Polar Form, Euler's Formula, and Complex Exponentials

Geometric Coordinate Conversion:
- Let point $P(a, b)$ represent $z = a + bi$ , at distance $r = |z|$ from the origin making angle $\theta$ with the positive real axis: $a = r\cos(\theta)$ $b = r\sin(\theta)$ $r = \sqrt{a^2 + b^2}$ $\tan(\theta) = \frac{b}{a} \implies \theta = \tan^{-1}\left(\frac{b}{a}\right)$
Definition of Argument:
- The argument $\theta = \arg(z)$ is the angle between the positive real axis and the position vector of the complex number.
Polar Form Definition:
- The polar representation of a complex number is: $z = r(\cos(\theta) + i\sin(\theta))$

Multiplication and Division in Polar Form:
- Multiplication Rule: Multiply moduli, add arguments: $z_1 z_2 = r_1 r_2 [(\cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)) + i(\sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi))]$ $z_1 z_2 = r_1 r_2 [\cos(\theta + \phi) + i\sin(\theta + \phi)]$
- Example: If $z_1 = 3(\cos(30^\circ) + i\sin(30^\circ))$ and $z_2 = 4(\cos(60^\circ) + i\sin(60^\circ))$ : $z_1 z_2 = 3 \times 4 [\cos(30^\circ + 60^\circ) + i\sin(30^\circ + 60^\circ)] = 12[\cos(90^\circ) + i\sin(90^\circ)] = 12i$
- Division Rule: Divide moduli, subtract argument of denominator from numerator: $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta - \phi) + i\sin(\theta - \phi)]$
Euler's Formula:
- For any real number $\theta$ : $e^{i\theta} = \cos(\theta) + i\sin(\theta)$
Exponential Form of a Complex Number:
- Expressed as: $z = re^{i\theta}$
- Modulus verification: $|z| = |re^{i\theta}| = |r| \cdot |\cos(\theta) + i\sin(\theta)| = r\sqrt{\cos^2(\theta) + \sin^2(\theta)} = r$
- Since $z = r(\cos(\theta) + i\sin(\theta))$ traces a circle of radius $r$ , $z = re^{i\theta}$ serves as a parametric representation of a circle of radius $r$ .

De Moivre’s Theorem and Complex Roots

De Moivre's Theorem Statement:
- If $z = r(\cos(\theta) + i\sin(\theta))$ and $n$ is a positive integer, then: $z^n = [r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$
- Powers of a complex number are obtained by raising the modulus to the $n$ -th power and multiplying the argument by $n$ .
Simplification Examples:
- Simplify $(\cos(\theta) - i\sin(\theta))^4$ : $(\cos(\theta) - i\sin(\theta))^4 = (\cos(-\theta) + i\sin(-\theta))^4 = \cos(-4\theta) + i\sin(-4\theta) = \cos(4\theta) - i\sin(4\theta)$
- Simplify $(\sin(\theta) - i\cos(\theta))^7$ : $(\sin(\theta) - i\cos(\theta))^7 = [-i(\cos(\theta) + i\sin(\theta))]^7 = (-i)^7(\cos(\theta) + i\sin(\theta))^7 = i(\cos(7\theta) + i\sin(7\theta)) = i\cos(7\theta) - \sin(7\theta)$
- Simplify $\frac{(\cos(2\theta) + i\sin(2\theta))^3}{(\cos(2\theta) - i\sin(2\theta))^4}$ : $\frac{\cos(6\theta) + i\sin(6\theta)}{\cos(-8\theta) + i\sin(-8\theta)} = \cos(6\theta - (-8\theta)) + i\sin(6\theta - (-8\theta)) = \cos(14\theta) + i\sin(14\theta)$
Expressing Powers in $x + iy$ Form:
- Example 1: Evaluate $\left[\sqrt{2}\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right)\right]^6$ : $(\sqrt{2})^6 \left(\cos\left(\frac{6\pi}{2}\right) + i\sin\left(\frac{6\pi}{2}\right)\right) = 8(\cos(3\pi) + i\sin(3\pi)) = 8(-1 + 0i) = -8$
- Example 2: Evaluate $(1 + \sqrt{3}i)^{10}$ :
- Modulus: $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$ .
- Argument: $\theta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$ .
- In polar form: $1 + \sqrt{3}i = 2\left(\cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right)\right)$ .
- Applying theorem: $2^{10}\left(\cos\left(\frac{10\pi}{3}\right) + i\sin\left(\frac{10\pi}{3}\right)\right) = 1024\left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right) = -512 - 512\sqrt{3}i$
- Example 3: Evaluate $\left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i\right)^{10}$ :
- Polar form: $r = 1$ , $\theta = \frac{\pi}{4}$ .
- Result: $\cos\left(\frac{10\pi}{4}\right) + i\sin\left(\frac{10\pi}{4}\right) = \cos\left(\frac{5\pi}{2}\right) + i\sin\left(\frac{5\pi}{2}\right) = 0 + i = i$ .
Roots of a Complex Number:
- For $z = r(\cos(\theta) + i\sin(\theta))$ and integer $n \ge 1$ , there are $n$ distinct $n$ -th roots given by: $w_k = r^{1/n}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right], \quad k = 0, 1, 2, \dots, n - 1$
- Geometric distribution: All $n$ roots lie equally spaced on a circle of radius $r^{1/n}$ centered at the origin, with angular separation $\frac{2\pi}{n}$ .
- Worked Example: Find the sixth roots of $z = -8$ :
- Polar form of $-8$ : $r = 8$ , $\theta = \pi$ , so $z = 8(\cos(\pi) + i\sin(\pi))$ .
- Sixth roots have radius $r^{1/6} = 8^{1/6} = (2^3)^{1/6} = 2^{1/2} = \sqrt{2}$ .
- General formula: $w_k = \sqrt{2}\left[\cos\left(\frac{\pi + 2k\pi}{6}\right) + i\sin\left(\frac{\pi + 2k\pi}{6}\right)\right]$ for $k = 0, 1, 2, 3, 4, 5$ .
- Evaluating roots:
  - $k = 0$ : $w_0 = \sqrt{2}\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right) = \sqrt{2}\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}i$
  - $k = 1$ : $w_1 = \sqrt{2}\left(\cos\left(\frac{3\pi}{6}\right) + i\sin\left(\frac{3\pi}{6}\right)\right) = \sqrt{2}\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right) = \sqrt{2}i$
  - $k = 2$ : $w_2 = \sqrt{2}\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right) = \sqrt{2}\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = -\frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}i$
  - $k = 3$ : $w_3 = \sqrt{2}\left(\cos\left(\frac{7\pi}{6}\right) + i\sin\left(\frac{7\pi}{6}\right)\right) = \sqrt{2}\left(-\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) = -\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}i$
  - $k = 4$ : $w_4 = \sqrt{2}\left(\cos\left(\frac{9\pi}{6}\right) + i\sin\left(\frac{9\pi}{6}\right)\right) = \sqrt{2}\left(\cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right)\right) = -\sqrt{2}i$
  - $k = 5$ : $w_5 = \sqrt{2}\left(\cos\left(\frac{11\pi}{6}\right) + i\sin\left(\frac{11\pi}{6}\right)\right) = \sqrt{2}\left(\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) = \frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}i$

Applications of De Moivre’s Theorem

Summation of Trigonometric Series:
- Consider the series: $C = \sum_{r=1}^n \cos(r\theta) = \cos(\theta) + \cos(2\theta) + \dots + \cos(n\theta)$ $S = \sum_{r=1}^n \sin(r\theta) = \sin(\theta) + \sin(2\theta) + \dots + \sin(n\theta)$
- Let $z = \cos(\theta) + i\sin(\theta)$ . By De Moivre's theorem, $z^r = \cos(r\theta) + i\sin(r\theta)$ .
- Express combined sum as a geometric progression: $C + iS = \sum_{r=1}^n z^r = z + z^2 + \dots + z^n = \frac{z(1 - z^n)}{1 - z}$
- Algebraic manipulation of $1 - z^n$ and $1 - z$ : $1 - z = 1 - \cos(\theta) - i\sin(\theta) = 2\sin^2\left(\frac{\theta}{2}\right) - 2i\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right) = -2i\sin\left(\frac{\theta}{2}\right)\left[\cos\left(\frac{\theta}{2}\right) + i\sin\left(\frac{\theta}{2}\right)\right]$ $1 - z^n = -2i\sin\left(\frac{n\theta}{2}\right)\left[\cos\left(\frac{n\theta}{2}\right) + i\sin\left(\frac{n\theta}{2}\right)\right]$
- Dividing the expressions and equating real and imaginary parts yields: $\sum_{r=1}^n \cos(r\theta) = \frac{\cos\left(\frac{n+1}{2}\theta\right)\sin\left(\frac{n\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)}$ $\sum_{r=1}^n \sin(r\theta) = \frac{\sin\left(\frac{n+1}{2}\theta\right)\sin\left(\frac{n\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)}$
Multiple Angle Polynomial Expansions:
- Using $\cos(n\theta) + i\sin(n\theta) = (\cos(\theta) + i\sin(\theta))^n$ , expand the right side via the Binomial Theorem and equate real and imaginary parts.
- Worked Example: Expand $\cos(3\theta)$ and $\sin(3\theta)$ :
- Let $c = \cos(\theta)$ and $s = \sin(\theta)$ .
- Expansion: $(\cos(\theta) + i\sin(\theta))^3 = c^3 + 3c^2(is) + 3c(is)^2 + (is)^3 = c^3 + 3ic^2s - 3cs^2 - is^3 = (c^3 - 3cs^2) + i(3c^2s - s^3)$
- Real part ( $\cos(3\theta)$ ): $\cos(3\theta) = c^3 - 3cs^2 = \cos^3(\theta) - 3\cos(\theta)\sin^2(\theta) = \cos^3(\theta) - 3\cos(\theta)(1 - \cos^2(\theta)) = 4\cos^3(\theta) - 3\cos(\theta)$
- Imaginary part ( $\sin(3\theta)$ ): $\sin(3\theta) = 3c^2s - s^3 = 3(1 - \sin^2(\theta))\sin(\theta) - \sin^3(\theta) = 3\sin(\theta) - 4\sin^3(\theta)$

Matrix Fundamentals and Special Matrices

Matrix Definition and Dimensions:
- A matrix is a rectangular array of elements arranged in rows and columns.
- An $m \times n$ matrix contains $m$ horizontal rows and $n$ vertical columns.
- Matrix element notation: $a_{ij}$ specifies the entry in row $i$ and column $j$ .
Classification of Special Matrices:
- Square Matrix: A matrix with an equal number of rows and columns ( $m = n$ ).
- Principal Diagonal: The entries $a_{11}, a_{22}, \dots, a_{nn}$ of a square matrix.
- Trace of a Matrix: The scalar sum of all elements along the principal diagonal: $\text{tr}(A) = \sum_{i=1}^n a_{ii}$
- Example: For $A = \begin{pmatrix} 3 & 6 & 9 \\ 12 & 15 & 18 \\ 21 & 24 & 27 \end{pmatrix}$ , $\text{tr}(A) = 3 + 15 + 27 = 45$ .
- Sub-Matrix: Any matrix obtained by deleting a subset of rows and/or columns from a given matrix.
- Zero Matrix ( $0$ ): A matrix in which every entry is zero.
- Identity Matrix ( $I$ ): A square matrix with ones along the principal diagonal and zeros elsewhere: $I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
- Transpose of a Matrix ( $A^T$ ): Obtained by exchanging rows with columns ( $(A^T)_{ij} = A_{ji}$ ).
- Conjugate Transpose ( $A^*$ or $A^H$ ): For a complex matrix, obtained by transposing and taking the complex conjugate of each entry ( $(A^*)_{ij} = \overline{a_{ji}}$ ).
- Example: If $A = \begin{pmatrix} 4 & 5 + 3i \\ 2 - i & 2 \end{pmatrix}$ , then $A^* = \begin{pmatrix} 4 & 2 + i \\ 5 - 3i & 2 \end{pmatrix}$ .
- Symmetric Matrix: A square matrix identical to its transpose: $A = A^T$ ( $a_{ij} = a_{ji}$ for all $i, j$ ).
- Diagonal Matrix: A matrix whose off-diagonal entries are all zero ( $a_{ij} = 0$ for $i \neq j$ ).
Triangular Matrices:
- Upper Triangular Matrix: All entries below the principal diagonal are zero ( $a_{ij} = 0$ for $i > j$ ).
- Lower Triangular Matrix: All entries above the principal diagonal are zero ( $a_{ij} = 0$ for $i < j$ ).

Matrix Algebra and Operational Properties

Matrix Addition and Subtraction:
- Matrices must share identical dimensions ( $m \times n$ ).
- Operations occur entrywise: $(A \pm B)_{ij} = a_{ij} \pm b_{ij}$
- Properties of Matrix Addition:
- Commutative: $A + B = B + A$
- Associative: $(A + B) + C = A + (B + C)$
- Additive Identity: $A + 0 = A$
- Additive Inverse: $A + (-A) = 0$
- Scalar Distribution: $k(tA) = (kt)A$ , $(k + t)A = kA + tA$ , $k(A + B) = kA + kB$
Scalar Multiplication:
- Multiplying a matrix by a scalar $k$ scales each entry: $(kA)_{ij} = k \cdot a_{ij}$
Matrix Multiplication:
- Inner Dimension Compatibility: The product $AB$ is defined if and only if the number of columns in $A$ equals the number of rows in $B$ .
- If $A$ is of size $m \times n$ and $B$ is of size $n \times k$ , the product $C = AB$ is of size $m \times k$ with entries: $c_{ij} = \sum_{r=1}^n a_{ir} b_{rj}$
- Worked Example: Compute $AB$ for $A = \begin{pmatrix} 0 & 1 & 2 \\ 3 & 4 & 5 \end{pmatrix}$ ( $2 \times 3$ ) and $B = \begin{pmatrix} 6 & 7 \\ 8 & 9 \\ 10 & 11 \end{pmatrix}$ ( $3 \times 2$ ): $c_{11} = 0(6) + 1(8) + 2(10) = 28$ $c_{12} = 0(7) + 1(9) + 2(11) = 31$ $c_{21} = 3(6) + 4(8) + 5(10) = 18 + 32 + 50 = 100$ $c_{22} = 3(7) + 4(9) + 5(11) = 21 + 36 + 55 = 112$ $AB = \begin{pmatrix} 28 & 31 \\ 100 & 112 \end{pmatrix}$
Properties of Matrix Multiplication:
- Non-Commutative: In general, $AB \neq BA$ .
Multiplication Order Terminology:
- In $AB$ , $A$ is post-multiplied by $B$ , or $B$ is pre-multiplied by $A$ .
Associative Property: $A(BC) = (AB)C$
Distributive Properties: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
Scalar Compatibility: $k(AB) = (kA)B = A(kB)$
Multiplicative Identity: For any square matrix $A$ , $IA = AI = A$ .

Determinants, Adjoints, and Inverses

Determinant of a Matrix:
- The determinant is a scalar value associated with square matrices, representing the oriented volume scaling factor of the linear transformation.
- For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ : $\det(A) = |A| = ad - bc$
Fundamental Properties of Determinants:
- If two rows or columns are equal or linearly dependent, $\det(A) = 0$ .
- Invariance under transpose: $\det(A^T) = \det(A)$ .
- Multiplicative property: $\det(AB) = \det(A)\det(B)$ .
- Triangular matrices: The determinant of an upper or lower triangular matrix is the product of its principal diagonal elements: $\det(A) = \prod_{i=1}^n a_{ii}$
Minors and Cofactors:
- Minor ( $M_{ij}$
- The determinant of the $(n - 1) \times (n - 1)$ sub-matrix formed by deleting the $i$ -th row and $j$ -th column of $A$ .
- Cofactor ( $C_{ij}$
- The signed minor given by: $C_{ij} = (-1)^{i+j} M_{ij}$
- Sign pattern forms an alternating checkerboard: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$ .
- Laplace Expansion: The determinant of an $n \times n$ matrix along the $i$ -th row is: $\det(A) = \sum_{j=1}^n a_{ij} C_{ij}$
Adjoint (Adjugate) of a Matrix ( $\text{Adj}(A)$
- The transpose of the cofactor matrix $C$ : $\text{Adj}(A) = C^T$
Matrix Inversion:
- The inverse of an $n \times n$ matrix $A$ is the unique matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ .
- Existence: $A^{-1}$ exists if and only if $\det(A) \neq 0$ (non-singular matrix).
- Formula via Adjoint: $A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)$
Comprehensive Worked Example of Inversion:
- Let $A = \begin{pmatrix} 2 & 3 & 5 \\ 4 & 1 & 6 \\ 1 & 4 & 0 \end{pmatrix}$ .
- Compute Minors: $M_{11} = \det\begin{pmatrix} 1 & 6 \\ 4 & 0 \end{pmatrix} = -24, \quad M_{12} = \det\begin{pmatrix} 4 & 6 \\ 1 & 0 \end{pmatrix} = -6, \quad M_{13} = \det\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} = 15$ $M_{21} = \det\begin{pmatrix} 3 & 5 \\ 4 & 0 \end{pmatrix} = -20, \quad M_{22} = \det\begin{pmatrix} 2 & 5 \\ 1 & 0 \end{pmatrix} = -5, \quad M_{23} = \det\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} = 5$ $M_{31} = \det\begin{pmatrix} 3 & 5 \\ 1 & 6 \end{pmatrix} = 13, \quad M_{32} = \det\begin{pmatrix} 2 & 5 \\ 4 & 6 \end{pmatrix} = -8, \quad M_{33} = \det\begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix} = -10$
- Form Cofactor Matrix $C$ : $C = \begin{pmatrix} -24 & 6 & 15 \\ 20 & -5 & -5 \\ 13 & 8 & -10 \end{pmatrix}$
- Transpose to find Adjoint $\text{Adj}(A) = C^T$ : $\text{Adj}(A) = \begin{pmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{pmatrix}$
- Compute Determinant along first row: $\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} = 2(-24) + 3(6) + 5(15) = -48 + 18 + 75 = 45$
- Compute Inverse: $A^{-1} = \frac{1}{45}\begin{pmatrix} -24 & 20 & 13 \\ 6 & -5 & 8 \\ 15 & -5 & -10 \end{pmatrix} = \begin{pmatrix} -\frac{8}{15} & \frac{4}{9} & \frac{13}{45} \\ \frac{2}{15} & -\frac{1}{9} & \frac{8}{45} \\ \frac{1}{3} & -\frac{1}{9} & -\frac{2}{9} \end{pmatrix}$

Row Echelon Forms, Vector Independence, and Rank

Row Echelon Form (REF):
- Conditions for REF:
- The first non-zero element in each non-zero row (leading entry or pivot) is 1.
- Each leading entry is located in a column strictly to the right of the leading entry in the preceding row.
- Any rows consisting entirely of zeros are grouped at the bottom.
Reduced Row Echelon Form (RREF):
- Conditions for RREF:
- Satisfies all conditions of Row Echelon Form.
- Each column containing a leading 1 has zeros everywhere else in that column.
- Uniqueness: While multiple REF matrices can represent a given matrix, the RREF is strictly unique.
Elementary Row Operations:
- Type I: Interchanging two rows ( $R_i \leftrightarrow R_j$ ).
- Type II: Multiplying a row by a non-zero scalar ( $R_i \to kR_i$ ).
- Type III: Adding a scalar multiple of one row to another ( $R_i \to R_i + kR_j$ ).
Linear Combinations and Vector Independence:
- A vector $\mathbf{v}$ is a linear combination of vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_k$ if scalars $c_1, c_2, \dots, c_k$ exist such that: $\mathbf{v} = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \dots + c_k\mathbf{u}_k$
- Linear Independence: A set of vectors is linearly independent if: $c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \dots + c_k\mathbf{u}_k = \mathbf{0} \implies c_1 = c_2 = \dots = c_k = 0$
- Linear Dependence: Holds if at least one vector can be written as a linear combination or scalar multiple of the others.
Rank of a Matrix:
- Definition 1 (Determinantal Rank): The size $r$ of the largest square sub-matrix with a non-zero determinant.
- Definition 2 (Row/Column Rank): The maximum number of linearly independent row vectors (or column vectors) in the matrix, equivalent to the number of non-zero rows in its Row Echelon Form.
- Upper Bound: For an $r \times c$ matrix, $\text{rank}(A) \le \min(r, c)$ .
- Full Rank: An $m \times n$ matrix is full rank if $\text{rank}(A) = \min(m, n)$ .

Systems of Linear Equations

Linear System Representation:
- A system of $n$ equations in $n$ unknowns: $a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = y_1$ $a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = y_2$ $\vdots$ $a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n = y_n$
- Matrix equation form: $Ax = y$ , where $A$ is the $n \times n$ coefficient matrix, $x$ is the column vector of unknowns, and $y$ is the column vector of constants.
- Homogeneous Systems: If $y = \mathbf{0}$ , the system is homogeneous; otherwise, it is non-homogeneous.
Method 1: Matrix Inversion Method
- Multiply both sides by $A^{-1}$ : $A^{-1}(Ax) = A^{-1}y \implies Ix = A^{-1}y \implies x = A^{-1}y$
- Worked Example: Solve the system: $x_1 + 2x_2 + 2x_3 = 1$ $2x_1 + 2x_2 + 2x_3 = 2$ $2x_1 + 2x_2 + x_3 = 3$
- Write in matrix form $Ax = y$ : $\begin{pmatrix} 1 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$
- Compute $A^{-1}$ : $A^{-1} = \begin{pmatrix} -1 & 1 & 0 \\ 1 & -1.5 & 1 \\ 0 & 1 & -1 \end{pmatrix}$
- Compute solution vector: $\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} -1 & 1 & 0 \\ 1 & -1.5 & 1 \\ 0 & 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} -1(1) + 1(2) + 0(3) \\ 1(1) - 1.5(2) + 1(3) \\ 0(1) + 1(2) - 1(3) \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$ $x_1 = 1, \quad x_2 = 1, \quad x_3 = -1$
Method 2: Cramer’s Rule
- Determinant evaluation of individual variables: Let $D = \det(A)$ . Let $D_{x_j}$ be the determinant formed by replacing the $j$ -th column of $A$ with vector $y$ .
- Variable formula: $x_j = \frac{D_{x_j}}{D}$
- Evaluation for the above system:
- Coefficient Determinant: $D = \det\begin{pmatrix} 1 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 1 \end{pmatrix} = 2$
- $D_{x_1} = \det\begin{pmatrix} 1 & 2 & 2 \\ 2 & 2 & 2 \\ 3 & 2 & 1 \end{pmatrix} = 2 \implies x_1 = \frac{2}{2} = 1$
- $D_{x_2} = \det\begin{pmatrix} 1 & 1 & 2 \\ 2 & 2 & 2 \\ 2 & 3 & 1 \end{pmatrix} = 2 \implies x_2 = \frac{2}{2} = 1$
- $D_{x_3} = \det\begin{pmatrix} 1 & 2 & 1 \\ 2 & 2 & 2 \\ 2 & 2 & 3 \end{pmatrix} = -2 \implies x_3 = \frac{-2}{2} = -1$
Method 3: Gaussian Elimination
- Form augmented matrix: $[A | y]$ .
- Perform elementary row operations to reduce $A$ to an upper triangular form.
- Augmented Matrix Reduction: $\begin{pmatrix} 1 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 \\ 2 & 2 & 1 & 3 \end{pmatrix}$
- Row operations: $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 2R_1$ : $\begin{pmatrix} 1 & 2 & 2 & 1 \\ 0 & -2 & -2 & 0 \\ 0 & -2 & -3 & 1 \end{pmatrix}$
- Scale row 2 ( $R_2 \to -\frac{1}{2}R_2$ ): $\begin{pmatrix} 1 & 2 & 2 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & -2 & -3 & 1 \end{pmatrix}$
- Eliminate entry in row 3 ( $R_3 \to R_3 + 2R_2$ ): $\begin{pmatrix} 1 & 2 & 2 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & -1 & 1 \end{pmatrix}$
- Scale row 3 ( $R_3 \to -R_3$ ): $\begin{pmatrix} 1 & 2 & 2 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 \end{pmatrix}$
- Back-Substitution:
- From row 3: $x_3 = -1$
- From row 2: $x_2 + x_3 = 0 \implies x_2 - 1 = 0 \implies x_2 = 1$
- From row 1: $x_1 + 2(1) + 2(-1) = 1 \implies x_1 = 1$

Eigenvalues and Eigenvectors

Fundamental Definition:
- Let $A$ be an $n \times n$ matrix. A scalar $\lambda$ is an eigenvalue of $A$ if there exists a non-zero vector $\mathbf{v}$ such that: $A\mathbf{v} = \lambda \mathbf{v}$
- Vector $\mathbf{v}$ is called an eigenvector corresponding to eigenvalue $\lambda$ .
Characteristic Equation and Polynomial:
- Rewrite eigenvalue equation as: $(A - \lambda I)\mathbf{v} = \mathbf{0}$
- For non-trivial solutions ( $\mathbf{v} \neq \mathbf{0}$ ), the matrix $(A - \lambda I)$ must be singular: $p(\lambda) = \det(A - \lambda I) = 0$
- $p(\lambda)$ is the characteristic polynomial of $A$ , and its roots are the eigenvalues.
Worked Example:
- Find eigenvalues and eigenvectors of $A = \begin{pmatrix} 2 & -4 \\ -1 & -1 \end{pmatrix}$ :
- Characteristic Determinant: $\det(A - \lambda I) = \det\begin{pmatrix} 2 - \lambda & -4 \\ -1 & -1 - \lambda \end{pmatrix} = (2 - \lambda)(-1 - \lambda) - (-4)(-1)$ $p(\lambda) = (\lambda^2 - \lambda - 2) - 4 = \lambda^2 - \lambda - 6 = (\lambda - 3)(\lambda + 2) = 0$
- Eigenvalues: $\lambda_1 = 3$ and $\lambda_2 = -2$ .
- Finding Eigenvector for $\lambda_1 = 3$ : $\begin{pmatrix} 2 - 3 & -4 \\ -1 & -1 - 3 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} -1 & -4 \\ -1 & -4 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ $-v_1 - 4v_2 = 0 \implies v_1 = -4v_2$
- Setting $v_2 = 1$ yields eigenvector $\mathbf{v}_1 = \begin{pmatrix} -4 \\ 1 \end{pmatrix}$ .
- Finding Eigenvector for $\lambda_2 = -2$ : $\begin{pmatrix} 2 - (-2) & -4 \\ -1 & -1 - (-2) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 4 & -4 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ $-v_1 + v_2 = 0 \implies v_1 = v_2$
- Setting $v_2 = 1$ yields eigenvector $\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ .

Vector Foundations and Vector Algebra

Vector Definition and Representation:
- A vector is a quantity characterized by both magnitude and direction.
- Directed line segment representation: $\vec{PQ}$ has initial point $P$ and terminal point $Q$ .
- Scalar quantities: Characterized solely by magnitude (e.g., mass, distance, speed, temperature, energy).
- Vector quantities: Require magnitude and direction (e.g., velocity, acceleration, force, momentum).
Classifications of Vectors:
- Zero Vector (Null Vector, $\mathbf{0}$ ): Magnitude of zero with undefined or arbitrary direction.
- Unit Vector ( $\hat{\mathbf{a}}$
- Magnitude of 1 unit: $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$
- Equal Vectors: Possess identical magnitude and direction regardless of spatial origin.
- Collinear Vectors: Vectors lying along the same straight line or parallel lines.
- Concurrent Vectors: Vectors whose lines of action intersect at a common point.
- Coplanar Vectors: Vectors situated within the same geometric plane.
Laws of Vector Algebra:
- Commutative Law of Addition: $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$
- Associative Law of Addition: $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$
- Associative Law of Scalar Multiplication: $m(n\mathbf{a}) = (mn)\mathbf{a}$
- Distributive Laws: $(m + n)\mathbf{a} = m\mathbf{a} + n\mathbf{a}$ and $m(\mathbf{a} + \mathbf{b}) = m\mathbf{a} + m\mathbf{b}$
Geometric Addition Principles:
- Parallelogram Law: If two vectors $\mathbf{a}$ and $\mathbf{b}$ originate from a common point, forming adjacent sides of a parallelogram, their resultant $\mathbf{a} + \mathbf{b}$ is the directed diagonal originating from that common point.
- Triangle Law: If two vectors $\mathbf{a}$ and $\mathbf{b}$ are arranged head-to-tail, their resultant is the directed segment completing the triangle from the initial point of the first to the terminal point of the second.
- Polygon Law: For $n-1$ vectors represented head-to-tail along an $n$ -sided polygon, their resultant is the closing side drawn in reverse order.
Position Vectors and Midpoints:
- The position vector of point $A$ relative to origin $O$ is $\vec{OA} = \mathbf{a}$ .
- Displacement vector between points $A$ and $B$ : $\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{b} - \mathbf{a}$
- Midpoint Position Vector: If $M$ bisects $AB$ : $\vec{OM} = \vec{OA} + \frac{1}{2}\vec{AB} = \mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) = \frac{1}{2}(\mathbf{a} + \mathbf{b})$

Coordinate Vectors and Direction Cosines

Component Representation in Three Dimensions:
- Using a right-handed orthogonal system with unit basis vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$ along the $x, y, z$ axes: $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$
- Magnitude of Vector $\mathbf{a}$ : $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
- Example: Given $\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 4\mathbf{k}$ , $\mathbf{b} = 5\mathbf{j} + 2\mathbf{k}$ , $\mathbf{c} = \mathbf{i} - 2\mathbf{j} - \mathbf{k}$ :
- Resultant: $\mathbf{a} + \mathbf{b} + \mathbf{c} = (3 + 0 + 1)\mathbf{i} + (-1 + 5 - 2)\mathbf{j} + (4 + 2 - 1)\mathbf{k} = 4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}$
- Magnitude of $\mathbf{a} - \mathbf{b}$ : $\mathbf{a} - \mathbf{b} = 3\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}$ , so $|\mathbf{a} - \mathbf{b}| = \sqrt{3^2 + (-6)^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7$ .
Direction Angles and Direction Cosines:
- The direction angles $\alpha, \beta, \gamma \in [0, \pi]$ are the angles between non-zero vector $\mathbf{a}$ and the positive $x, y, z$ coordinate axes.
- Direction Cosines: Denoted by $\ell, \mu, \rho$ : $\ell = \cos(\alpha) = \frac{a_1}{|\mathbf{a}|}$ $\mu = \cos(\beta) = \frac{a_2}{|\mathbf{a}|}$ $\rho = \cos(\gamma) = \frac{a_3}{|\mathbf{a}|}$
- Fundamental Theorem: The sum of the squares of the direction cosines equals 1: $\ell^2 + \mu^2 + \rho^2 = \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$
- Proof: $\ell^2 + \mu^2 + \rho^2 = \left(\frac{a_1}{|\mathbf{a}|}\right)^2 + \left(\frac{a_2}{|\mathbf{a}|}\right)^2 + \left(\frac{a_3}{|\mathbf{a}|}\right)^2 = \frac{a_1^2 + a_2^2 + a_3^2}{|\mathbf{a}|^2} = \frac{|\mathbf{a}|^2}{|\mathbf{a}|^2} = 1$
- Unit Vector Relation: The components of a unit vector are its direction cosines: $\hat{\mathbf{a}} = \ell\mathbf{i} + \mu\mathbf{j} + \rho\mathbf{k}$
- Worked Example: Find direction cosines for $\mathbf{a} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ :
- Magnitude: $|\mathbf{a}| = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$ .
- Direction cosines: $\ell = \frac{1}{\sqrt{14}}, \quad \mu = -\frac{2}{\sqrt{14}}, \quad \rho = \frac{3}{\sqrt{14}}$
- Verification: $\left(\frac{1}{\sqrt{14}}\right)^2 + \left(-\frac{2}{\sqrt{14}}\right)^2 + \left(\frac{3}{\sqrt{14}}\right)^2 = \frac{1 + 4 + 9}{14} = 1$ .

Vector Multiplication: Dot and Cross Products

Scalar (Dot) Product:
- Definition: For vectors $\mathbf{a}$ and $\mathbf{b}$ with angle $\theta \in [0, \pi]$ between them: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)$
- Algebraic Component Form: $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$
- Properties of Dot Product:
- Commutative: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$
- Distributive: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$
- Basis Orthogonality: $\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1$ and $\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0$
- Perpendicularity Condition: Two non-zero vectors are perpendicular if and only if $\mathbf{a} \cdot \mathbf{b} = 0$ .
- Parallel Condition: If $\mathbf{a} \parallel \mathbf{b}$ , then $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|$
Physical and Geometric Applications of Dot Product:
- Angle Between Vectors: $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} = \frac{a_1 b_1 + a_2 b_2 + a_3 b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2}\sqrt{b_1^2 + b_2^2 + b_3^2}}$
- Distance of Point from Origin: $d = \sqrt{\mathbf{r} \cdot \mathbf{r}} = |\mathbf{r}|$
- Work Done by a Constant Force: $W = \mathbf{F} \cdot \mathbf{r} = |\mathbf{F}||\mathbf{r}|\cos(\theta)$
- Worked Example: A constant force of $5\,\text{N}$ in direction $\mathbf{i} + \mathbf{j} - 2\mathbf{k}$ displaces an object from $(1, 0, 1)$ to $(3, 4, -1)$ .
- Unit direction vector: $\hat{\mathbf{n}} = \frac{\mathbf{i} + \mathbf{j} - 2\mathbf{k}}{\sqrt{1^2 + 1^2 + (-2)^2}} = \frac{\mathbf{i} + \mathbf{j} - 2\mathbf{k}}{\sqrt{6}}$
- Force vector: $\mathbf{F} = 5\hat{\mathbf{n}} = \frac{5}{\sqrt{6}}(\mathbf{i} + \mathbf{j} - 2\mathbf{k})$
- Displacement vector: $\mathbf{r} = (3-1)\mathbf{i} + (4-0)\mathbf{j} + (-1-1)\mathbf{k} = 2\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$
- Work Done: $W = \mathbf{F} \cdot \mathbf{r} = \frac{5}{\sqrt{6}}[1(2) + 1(4) + (-2)(-2)] = \frac{5}{\sqrt{6}}[2 + 4 + 4] = \frac{50}{\sqrt{6}}\,\text{J}$
Vector (Cross) Product:
- Definition: Generates a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ : $\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin(\theta)\,\hat{\mathbf{n}}$
- Where $\hat{\mathbf{n}}$ is the unit normal vector specified by the right-hand rule.
- Properties of Cross Product:
- Anti-Commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$
- Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$
- Parallel Condition: $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ if and only if $\mathbf{a} \parallel \mathbf{b}$
- Cyclic Basis Properties: $\mathbf{i} \times \mathbf{i} = \mathbf{j} \times \mathbf{j} = \mathbf{k} \times \mathbf{k} = \mathbf{0}$ $\mathbf{i} \times \mathbf{j} = \mathbf{k}, \quad \mathbf{j} \times \mathbf{k} = \mathbf{i}, \quad \mathbf{k} \times \mathbf{i} = \mathbf{j}$ $\mathbf{j} \times \mathbf{i} = -\mathbf{k}, \quad \mathbf{k} \times \mathbf{j} = -\mathbf{i}, \quad \mathbf{i} \times \mathbf{k} = -\mathbf{j}$
- Determinant Representation: $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2)\mathbf{i} - (a_1 b_3 - a_3 b_1)\mathbf{j} + (a_1 b_2 - a_2 b_1)\mathbf{k}$
Geometric Applications of Cross Product:
- Area of Parallelogram with adjacent side vectors $\mathbf{a}$ and $\mathbf{b}$ : $\text{Area} = |\mathbf{a} \times \mathbf{b}|$
- Area of Triangle with adjacent side vectors $\mathbf{a}$ and $\mathbf{b}$ : $\text{Area} = \frac{1}{2}|\mathbf{a} \times \mathbf{b}|$
- Angle between vectors using cross product: $\sin(\theta) = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}||\mathbf{b}|}$

Fundamental Trigonometric Ratios and Evaluations

Right Triangle Trigonometric Ratios:
- For an acute angle $\theta$ ( $0^\circ < \theta < 90^\circ$ ) in a right triangle with adjacent side $a$ , opposite side $b$ , and hypotenuse $c$ : $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{b}{c}$ $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{c}$ $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{b}{a}$
Reciprocal Ratios:
- Defined as: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{c}{b}$ $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{c}{a}$ $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{a}{b}$
Geometric Derivation for Special Angles:
- Evaluation at $45^\circ$ ( $\frac{\pi}{4}$
- Constructed using an isosceles right triangle with side lengths $1, 1$ , giving hypotenuse $\sqrt{1^2 + 1^2} = \sqrt{2}$ . $\sin(45^\circ) = \frac{1}{\sqrt{2}}, \quad \cos(45^\circ) = \frac{1}{\sqrt{2}}, \quad \tan(45^\circ) = 1$ $\csc(45^\circ) = \sqrt{2}, \quad \sec(45^\circ) = \sqrt{2}, \quad \cot(45^\circ) = 1$
- Evaluation at $60^\circ$ ( $\frac{\pi}{3}$ ) and $30^\circ$ ( $\frac{\pi}{6}$
- Constructed by bisecting an equilateral triangle of side length 2, creating a right triangle with base 1, hypotenuse 2, and height $\sqrt{2^2 - 1^2} = \sqrt{3}$ .
- For $60^\circ$ : $\sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(60^\circ) = \frac{1}{2}, \quad \tan(60^\circ) = \sqrt{3}$ $\csc(60^\circ) = \frac{2}{\sqrt{3}}, \quad \sec(60^\circ) = 2, \quad \cot(60^\circ) = \frac{1}{\sqrt{3}}$
- For $30^\circ$ : $\sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \tan(30^\circ) = \frac{1}{\sqrt{3}}$ $\csc(30^\circ) = 2, \quad \sec(30^\circ) = \frac{2}{\sqrt{3}}, \quad \cot(30^\circ) = \sqrt{3}$
- Boundary Angle Evaluations ( $0^\circ$ and $90^\circ$
- At $0^\circ$ : $\sin(0^\circ) = 0$ , $\cos(0^\circ) = 1$ , $\tan(0^\circ) = 0$ , $\csc(0^\circ) = \infty$ , $\sec(0^\circ) = 1$ , $\cot(0^\circ) = \infty$ .
- At $90^\circ$ : $\sin(90^\circ) = 1$ , $\cos(90^\circ) = 0$ , $\tan(90^\circ) = \infty$ , $\csc(90^\circ) = 1$ , $\sec(90^\circ) = \infty$ , $\cot(90^\circ) = 0$ .

Analytical Trigonometric Identities

Pythagorean Identities: $\sin^2(\theta) + \cos^2(\theta) = 1$ $1 + \tan^2(\theta) = \sec^2(\theta)$ $1 + \cot^2(\theta) = \csc^2(\theta)$
Even-Odd Identities:
- Odd Functions ( $f(-x) = -f(x)$ $\sin(-\theta) = -\sin(\theta)$ $\tan(-\theta) = -\tan(\theta)$ $\csc(-\theta) = -\csc(\theta)$ $\cot(-\theta) = -\cot(\theta)$
- Even Functions ( $f(-x) = f(x)$ $\cos(-\theta) = \cos(\theta)$ $\sec(-\theta) = \sec(\theta)$
Quotient Identities: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$
Sum and Difference Identities:
- Cosine Identities: $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$
- Sine Identities: $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$
- Tangent Identities: $\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$ $\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$
Double Angle Identities:
- Double Angle for Sine: $\sin(2A) = 2\sin(A)\cos(A)$
- Double Angle for Cosine (Three Equivalent Forms): $\cos(2A) = \cos^2(A) - \sin^2(A)$ $\cos(2A) = 1 - 2\sin^2(A)$ $\cos(2A) = 2\cos^2(A) - 1$
- Double Angle for Tangent: $\tan(2A) = \frac{2\tan(A)}{1 - \tan^2(A)}$
Half Angle Identities:
- Derived by substituting $A = \frac{D}{2}$ into the double angle cosine formulas: $\cos\left(\frac{D}{2}\right) = \pm\sqrt{\frac{1 + \cos(D)}{2}}$ $\sin\left(\frac{D}{2}\right) = \pm\sqrt{\frac{1 - \cos(D)}{2}}$
Factor Formulas (Sum-to-Product Identities):
- Sum for Sine: $\sin(X) + \sin(Y) = 2\sin\left(\frac{X + Y}{2}\right)\cos\left(\frac{X - Y}{2}\right)$
- Sum for Cosine: $\cos(X) + \cos(Y) = 2\cos\left(\frac{X + Y}{2}\right)\cos\left(\frac{X - Y}{2}\right)$

Spherical Trigonometry

Core Definitions and Geometry of the Sphere:
- A spherical triangle is formed by the intersection of three great circle arcs on the surface of a sphere.
- Vertices are labeled $A, B, C$ with interior vertex angles $\alpha, \beta, \gamma$ .
- Sides are labeled $a, b, c$ , measured as angles subtended by the arcs at the sphere's center.

Angular Sum and Spherical Excess:
- Unlike planar triangles whose interior angle sum is exactly $\pi$ , the sum of angles of a spherical triangle satisfies: $\pi < \alpha + \beta + \gamma < 3\pi$
- Spherical Excess ( $E$ ): The positive angular quantity defined by: $E = \alpha + \beta + \gamma - \pi$
Surface Area of a Spherical Triangle:
- For a sphere of radius $R$ , with spherical excess $E$ measured in radians: $\text{Area} = R^2 E$
Right Spherical Triangles ( $\gamma = \frac{\pi}{2} = 90^\circ$
- The following fundamental identities govern right spherical triangles: $\sin(a) = \sin(\alpha)\sin(c) = \tan(b)\cot(\beta)$ $\sin(b) = \sin(\beta)\sin(c) = \tan(a)\cot(\alpha)$ $\cos(\alpha) = \cos(a)\sin(\beta) = \tan(b)\cot(c)$ $\cos(\beta) = \cos(b)\sin(\alpha) = \tan(a)\cot(c)$ $\cos(c) = \cot(\alpha)\cot(\beta) = \cos(a)\cos(b)$
General Spherical Triangles:
- Spherical Law of Sines: $\frac{\sin(a)}{\sin(\alpha)} = \frac{\sin(b)}{\sin(\beta)} = \frac{\sin(c)}{\sin(\gamma)}$
- Spherical Law of Cosines for Sides: $\cos(a) = \cos(b)\cos(c) + \sin(b)\sin(c)\cos(\alpha)$ $\cos(b) = \cos(a)\cos(c) + \sin(a)\sin(c)\cos(\beta)$ $\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(\gamma)$
- Spherical Law of Cosines for Angles: $\cos(\alpha) = -\cos(\beta)\cos(\gamma) + \sin(\beta)\sin(\gamma)\cos(a)$ $\cos(\beta) = -\cos(\alpha)\cos(\gamma) + \sin(\alpha)\sin(\gamma)\cos(b)$ $\cos(\gamma) = -\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\cos(c)$
Semi-Perimeter and Half-Angle Tangent Formulas:
- Let semi-perimeter of sides be $s = \frac{a + b + c}{2}$ . The vertex half-angle is given by: $\tan\left(\frac{\alpha}{2}\right) = \sqrt{\frac{\sin(s - b)\sin(s - c)}{\sin(s)\sin(s - a)}}$
- Let semi-sum of vertex angles be $\sigma = \frac{\alpha + \beta + \gamma}{2}$ . The side half-arc is given by: $\tan\left(\frac{a}{2}\right) = \sqrt{\frac{-\cos(\sigma)\cos(\sigma - \alpha)}{\cos(\sigma - \beta)\cos(\sigma - \gamma)}}$