Algebra and Vector Algebra Study Notes

Functions

A Cartesian product $A \times B$ is defined as the set $\{(a, b) \mid a \in A, b \in B\}$. A relation $f \subseteq A \times B$ is a function $f: A \rightarrow B$ if for every element $a \in A$, there exists a unique element $b \in B$ such that $f(a) = b$. The set $A$ is the domain, $B$ is the co-domain, and the set $f(A) = \{f(a) \mid a \in A\} \subseteq B$ is the range.

A function $f: A \rightarrow B$ is an injection (one-to-one) if $f(a_1) = f(a_2) \implies a_1 = a_2$ for all $a_1, a_2 \in A$. It is a surjection (onto) if the range $f(A) = B$. It is a bijection if it is both an injection and a surjection. If $f: A \rightarrow B$ is a bijection, its inverse $f^{-1}: B \rightarrow A$ exists as a bijection satisfying $f^{-1}(b) = a \iff f(a) = b$.

The composition of $f: A \rightarrow B$ and $g: B \rightarrow C$ is the function $(g \circ f): A \rightarrow C$ defined by $(g \circ f)(a) = g(f(a))$. If $f$ and $g$ are bijections, then $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$. A function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.

Mathematical Induction

The Well-Ordering Principle states that any non-empty set of positive integers contains a least element. This serves as the foundation for the Principle of Finite Mathematical Induction.

To prove a statement $P(n)$ for all natural numbers $n \in \mathbb{N}$, three core steps are executed:

1. Basis of Induction: Prove that $P(1)$ is true.

2. Inductive Hypothesis: Assume that $P(k)$ is true for an arbitrary positive integer $k \ge 1$

3. Inductive Step: Prove that $P(k+1)$ is true based on the inductive hypothesis $P(k)$.

Under Complete Mathematical Induction, the inductive hypothesis assumes that $P(1), P(2), \dots, P(k)$ are all true to show that $P(k+1)$ holds.

Matrices

A matrix is an ordered rectangular array of numbers or functions. A matrix with $m$ rows and $n$ columns has order $m \times n$. The trace of a square matrix $A$, denoted $\text{Tr}(A)$, is the sum of its principal diagonal elements $\sum_{i=1}^{n} a_{ii}$.

Matrix addition is commutative ($A + B = B + A$) and associative ($(A + B) + C = A + (B + C)$). Matrix multiplication $AB$ requires the number of columns of $A$ to equal the number of rows of $B$; it is associative ($(AB)C = A(BC)$) and distributive ($A(B+C) = AB + AC$), but generally non-commutative ($AB \neq BA$).

The transpose $A'$ satisfies $(A')' = A$, $(A + B)' = A' + B'$, and $(AB)' = B'A'$. A matrix $A$ is symmetric if $A' = A$ and skew-symmetric if $A' = -A$ (with diagonal elements equal to $0$). Every square matrix $A$ can be written uniquely as the sum of a symmetric matrix $\frac{1}{2}(A + A')$ and a skew-symmetric matrix $\frac{1}{2}(A - A')$.

A square matrix $A$ is non-singular if its determinant $\det(A) \neq 0$, and singular if $\det(A) = 0$. If $A$ is non-singular, its multiplicative inverse is $A^{-1} = \frac{\text{Adj}(A)}{\det(A)}$, where $\text{Adj}(A)$ is the transpose of the cofactor matrix of $A$. Inverse matrices satisfy $(AB)^{-1} = B^{-1}A^{-1}$ and $(A')^{-1} = (A^{-1})'$.

A system of linear equations $AX = D$ is consistent if $\text{rank}(A) = \text{rank}([A \mid D])$. It possesses a unique solution if $\text{rank}(A) = \text{rank}([A \mid D]) = n$, and infinitely many solutions if $\text{rank}(A) = \text{rank}([A \mid D]) < n$. If $\text{rank}(A) \neq \text{rank}([A \mid D])$, the system is inconsistent.

Addition of Vectors

A vector is a directed line segment possessing both magnitude and direction. A position vector $\mathbf{r} = \mathbf{OP} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ locates a point $P(x, y, z)$ relative to origin $O(0, 0, 0)$, with magnitude $|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$. The direction cosines $l = \cos(\alpha)$, $m = \cos(\beta)$, $n = \cos(\gamma)$ satisfy $l^2 + m^2 + n^2 = 1$.

Vector addition follows the Triangle Law ($\mathbf{AB} + \mathbf{BC} = \mathbf{AC}$) and the Parallelogram Law. Vector addition is commutative ($\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$) and associative ($(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$).

The position vector of a point $P$ dividing the segment joining $A(\mathbf{a})$ and $B(\mathbf{b})$ in the ratio $m : n$ is $\frac{m\mathbf{b} + n\mathbf{a}}{m + n}$ for internal division, and $\frac{m\mathbf{b} - n\mathbf{a}}{m - n}$ for external division. The centroid of a triangle with vertices $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is $\frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{3}$, and the centroid of a tetrahedron is $\frac{\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d}}{4}$.

The vector equation of a straight line through $\mathbf{a}$ parallel to $\mathbf{b}$ is $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. The line passing through two points $\mathbf{a}$ and $\mathbf{b}$ is $\mathbf{r} = (1-t)\mathbf{a} + t\mathbf{b}$. Three points $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are collinear if and only if $x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = \mathbf{0}$ with $x + y + z = 0$ for scalars $x, y, z$ not all zero.

Product of Vectors

The scalar (dot) product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)$. In component form, $\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$. Two non-zero vectors are perpendicular if and only if $\mathbf{a} \cdot \mathbf{b} = 0$. The orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ is $\frac{(\mathbf{a} \cdot \mathbf{b})\mathbf{a}}{|\mathbf{a}|^2}$, with magnitude $\frac{|\mathbf{a} \cdot \mathbf{b}|}{|\mathbf{a}|}$. The vector equation of a plane in normal form is $\mathbf{r} \cdot \mathbf{\hat{n}} = p$.

The vector (cross) product is $\mathbf{a} \times \mathbf{b} = (|\mathbf{a}||\mathbf{b}|\sin(\theta))\mathbf{\hat{n}}$, where $\mathbf{\hat{n}}$ forms a right-handed system with $\mathbf{a}$ and $\mathbf{b}$. Cross product is anti-commutative ($\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$). In component form, $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$. The vector area of a triangle with vertices $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is $\frac{1}{2}(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a})$, and the vector area of a parallelogram with adjacent sides $\mathbf{a}, \mathbf{b}$ is $\mathbf{a} \times \mathbf{b}$.

The scalar triple product is $[\mathbf{a}\;\mathbf{b}\;\mathbf{c}] = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}$. It satisfies cyclic symmetry $[\mathbf{a}\;\mathbf{b}\;\mathbf{c}] = [\mathbf{b}\;\mathbf{c}\;\mathbf{a}] = [\mathbf{c}\;\mathbf{a}\;\mathbf{b}]$. Non-zero vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are coplanar if and only if $[\mathbf{a}\;\mathbf{b}\;\mathbf{c}] = 0$. The volume of a parallelopiped with coterminus edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is $|[\mathbf{a}\;\mathbf{b}\;\mathbf{c}]|$, and the volume of a tetrahedron is \frac{1}{6}|[\mathbf{a}\;\mathbf{b}\;\mathbf{c}]|$.\n\nThe shortest distance between skew lines \mathbf{r} = \mathbf{a_1} + \lambda\mathbf{b_1}$and$\mathbf{r} = \mathbf{a_2} + \mu\mathbf{b_2}$is$d = \frac{|(\mathbf{a_2} - \mathbf{a_1}) \cdot (\mathbf{b_1} \times \mathbf{b_2})|}{|\mathbf{b_1} \times \mathbf{b_2}|}$. The vector triple product expansion is$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$$.