Introduction to Vector Algebra, Modular Systems, and Vector Calculus Properties

Review of Vector Basics and Linear Operations

• Vector Geometry in Two Dimensions ($2D$):     * Scaling: Multiplying a vector by a scalar changes its magnitude and direction. For example, if a vector is at $\begin{pmatrix} -2 \\ 4 \end{pmatrix}$, multiplying it by $2$ (to get $2v$) extends it twice in the same direction. Multiplying by $0.5$ halves its length. Multiplying by $-2$ extends its length twice but in the opposite direction.     * Vector Subtraction: Graphically, subtraction can be approached as adding the negative of a vector. For vectors $u$ and $v$, $u - v$ is equivalent to $u + (-v)$. This can be performed graphically using the parallelogram method or the tail-to-head method.

• Three-Dimensional Vectors ($3D$) and Beyond:     * Coordinates are represented as $(x, y, z)$.     * Example: A vector with components $[1, 2, 3]$ indicates moving $1$ unit in the positive $x$ direction, $2$ units in the positive $y$ direction, and $3$ units upward in the positive $z$ direction.     * Coordinate Systems: In $3D$, there are three planes. If all coordinates are positive, the vector resides in the first octant.     * $n$-Dimensional Space ($R^n$): Vectors can exist in any number of dimensions (e.g., $n = 10, 20, 30$).     * Notation: Vectors are usually written horizontally to save space, but they can be written vertically as well (column vectors). The definition of operations remains consistent regardless of orientation.

Algebraic Properties of Vectors

• Commutative Property of Addition:     * $u + v = v + u$     * Proof: Since vector addition is performed component-wise (e.g., $u_1 + v_1, u_2 + v_2$), and scalar addition is commutative (e.g., $2 + 3 = 3 + 2$), the final vector sum remains the same regardless of order.

• Associative Property of Addition:     * $(u + v) + w = u + (v + w)$     * Proof: When adding three vectors with $n$ components, the grouping of the components does not change the result: $(u_i + v_i) + w_i = u_i + (v_i + w_i)$.

• The Zero Vector ($\mathbf{0}$):     * Represented as a bold $\mathbf{0}$ or with an arrow notation $\vec{0}$. In $2D$, it is $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$; in $3D$, it is $\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$; and in $R^n$, it consists of $n$ zeros.     * Identity Property: $u + \mathbf{0} = u$.     * Additive Inverse: $u + (-u) = \mathbf{0}$.

• Distributive Properties:     * Scalar over vector sum: $c(u + v) = cu + cv$.     * Vector over scalar sum: $(c + d)u = cu + du$.

• Other Properties:     * $c(du) = (cd)u$     * $1 \cdot u = u$

Practical Examples: Simplification and Solving for Vectors

• Simplification Example:     * Simplify: $3(a + 5b - 2a) + 2(b - a)$.     * Step 1: Distribute and Combine inside parentheses: $3(-a + 5b) + 2(b - a)$.     * Step 2: Expand terms: $-3a + 15b + 2b - 2a$.     * Step 3: Combine like terms: $-5a + 17b$.     * Note: In the transcript, a similar problem results in $-4a + 7b$ or $-a+7b$ during a live walkthrough depending on distributive errors in the source slide, but the algebraic process of grouping like terms is the fundamental lesson.

• Solving for $x$ Example:     * Equation: $5(x - a) = 2(a + 2x)$.     * Step 1: Distribute the scalars: $5x - 5a = 2a + 4x$.     * Step 2: Subract $4x$ from both sides: $x - 5a = 2a$.     * Step 3: Add $5a$ to both sides: $x = 7a$.

Linear Combinations and Standard Vectors

• Linear Combination Definition:     * A vector $v$ is a linear combination of vectors $v_1, v_2, \dots, v_k$ if it can be written as:     * $v = c_1v_1 + c_2v_2 + \dots + c_kv_k$     * The constants $c_1, c_2, \dots, c_k$ are referred to as the coefficients.     * Example: Writing vector $w$ as a combination of $v_1$ and $v_2$ where $w = 2v_1 - 3v_2$.

• Standard Unit Vectors:     * In $2D$ ($R^2$):         * $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$         * $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$     * In $3D$ ($R^3$):         * $e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, $e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$     * Any vector can be written as a linear combination of standard vectors. For example, the point $(4, 2)$ is $4e_1 + 2e_2$.

Binary Systems and Modular Arithmetic ($Z_n$)

• Binary System ($Z_2$):     * Consists only of digits $0$ and $1$. This is the basis of computer operations and hardware (ON/OFF electricity).     * Addition in $Z_2$:         * $0 + 0 = 0$         * $0 + 1 = 1$         * $1 + 0 = 1$         * $1 + 1 = 0$ (Since $2$ is equivalent to $0$ in Modulo $2$).         * The additive inverse of $1$ is $1$.     * Multiplication in $Z_2$:         * $1 \times 1 = 1$         * Any multiplication by $0$ equals $0$.

• Integer Modulo $3$ ($Z_3$):     * Digits: ${0, 1, 2}$.     * Three-hour clock analogy: Adding $1$ to $2$ takes you back to $0$.     * Example Addition: $2 + 2 + 1 + 2 = 7$. In $Z_3$, $7 \div 3$ yields a remainder of $1$. Thus, the sum is $1$.     * Example Multiplication: $2 \times 2 = 4 \equiv 1 \pmod{3}$.

• Modular Vectors:     * Vectors can have components from modular systems. A binary vector of length $5$ ($Z_2^5$) contains five digits of either $0$ or $1$.     * Vector addition in modular systems is done component-wise.     * Example: $[1, 1, 0, 1, 0] + [0, 1, 1, 1, 0]$ in $Z_2$     * $1+0=1$; $1+1=0$; $0+1=1$; $1+1=0$; $0+0=0$. Result: $[1, 0, 1, 0, 0]$.

Dot Product (Scalar Product)

• Definition:     * For vectors $u$ and $v$ in $R^n$     * $u \cdot v = u_1v_1 + u_2v_2 + \dots + u_nv_n$     * The result of a dot product is always a scalar, not a vector.

• Properties of Dot Product:     * Commutative: $u \cdot v = v \cdot u$.     * Distributive: $u \cdot (v + w) = u \cdot v + u \cdot w$.     * Scalar Associativity: $c(u \cdot v) = (cu) \cdot v$.     * Positivity: $u \cdot u \ge 0$.     * $u \cdot u = 0$ if and only if $u = \mathbf{0}$.

Vector Norm and Length

• Length (Norm) Definition:     * The magnitude of a vector $v$, denoted as $\|v\|$, is the square root of the dot product of the vector with itself.     * $\|v\| = \sqrt{v \cdot v} = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}$     * Example: Length of vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ is $\sqrt{2^2 + 3^2} = \sqrt{13}$.

• Scalar Property of Norms:     * $\|cv\| = |c| \cdot \|v\|$. (The magnitude of a scalar times a vector is the absolute value of the scalar times the vector's length).

• Unit Vectors and Normalization:     * A Unit Vector is a vector with a length of exactly $1$.     * To normalize a vector $v$, divide the vector by its magnitude:     * $u = \frac{1}{\|v\|} v$     * Example: Normalizing $v = [2, -1, 3]$:         * $\|v\| = \sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{4+1+9} = \sqrt{14}$.         * Normalized vector $u = \begin{pmatrix} \frac{2}{\sqrt{14}} \\ -\frac{1}{\sqrt{14}} \\ \frac{3}{\sqrt{14}} \end{pmatrix}$.

Key Inequalities

• Cauchy-Schwarz Inequality:     * $|u \cdot v| \le \|u\| \cdot \|v\|$     * The absolute value of the dot product of two vectors is less than or equal to the product of their magnitudes.

• Triangle Inequality:     * $\|u + v\| \le \|u\| + \|v\|$     * The length of the sum of two vectors is less than or equal to the sum of the individual lengths of the two vectors. This corresponds to the geometric principle that any side of a triangle is shorter than the sum of the other two sides.

Distance, Angles, and Projections

• Distance Between Vectors:     * The distance $d(u, v)$ is defined as the magnitude of the difference vector:     * $d(u, v) = \|u - v\| = \sqrt{(u_1 - v_1)^2 + \dots + (u_n - v_n)^2}$.

• Angles and Orthogonality:     * Geometric Dot Product Formula: $u \cdot v = \|u\| \cdot \|v\| \cdot \cos(\theta)$.     * Rearranged for Angle Calculation: $\cos(\theta) = \frac{u \cdot v}{\|u\| \cdot \|v\|}$.     * Perpendicularity: Two vectors are perpendicular (orthogonal) if and only if their dot product is zero ($u \cdot v = 0$), because $\cos(90^\circ) = 0$.

• Projection:     * The projection of vector $v$ onto vector $u$ ($\text{proj}_u(v)$) is calculated as:     * $\text{proj}_u(v) = \frac{u \cdot v}{u \cdot u} u$     * Correction Note: The transcript notes a typo in the slide where the denominator was mislabeled, but clarifies the formula requires dividing by the dot product of the vector being projected onto (the base vector).

Questions & Discussion

• Attendance and Check-in: The instructor initiated a digital check-in through the faculty portal during a break.
• Pacing and Syllabus: Sections 1.1 and 1.2 were covered. Section 1.3 is scheduled for the following Monday. Students are encouraged to use WebAssign for homework for sections 1.1 and 1.2 and to keep up with the condensed summer schedule.
• Student Question on Absolute Value in Norms:     * Question: In the property $\|cv\| = |c|\|v\|$, if $c$ is negative, do we calculate it as positive?     * Response: Yes. Distance and length are never negative. Even if you walk in a negative direction on an axis, the physical distance traveled is positive. Therefore, the absolute value is applied to the scalar.