Comprehensive Guide to Unit 4: Vectors, Matrices, and Linear Transformations

Vector

A mathematical object possessing both magnitude (length) and direction.

Magnitude

The length of a vector, calculated using the Pythagorean theorem.

Direction Angle ($\theta$)

The angle a vector makes with the positive x-axis.

Standard Form of a Vector

Representation of a vector as $\vec{v} = \langle a, b \rangle$, where $a$ is the horizontal component and $b$ is the vertical component.

Unit Basis Vectors

Vectors defined as $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0, 1 \rangle$.

Vector Addition

The operation of combining two vectors, defined as $\vec{u} + \vec{v} = \langle u<em>1 + v</em>1, u<em>2 + v</em>2 \rangle$.

Scalar Multiplication

An operation that scales a vector's length by a scalar constant, expressed as $k\vec{v} = \langle ka, kb \rangle$.

Vector-Valued Function

A function that takes a scalar input and outputs a vector, typically expressed as $\vec{r}(t) = \langle f(t), g(t) \rangle$.

Parametric Equations

Equations that express $x$ and $y$ coordinates separately, often related to vector-valued functions.

Matrix

A rectangular array of numbers arranged in rows and columns.

Matrix Dimensions

The size of a matrix represented as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

Matrix Addition

The operation of adding two matrices, requiring the same dimensions.

Scalar Multiplication of a Matrix

The operation that multiplies every entry in the matrix by a scalar constant.

Matrix Multiplication Condition

To multiply Matrix $A$ ($m \times n$) by Matrix $B$ ($n \times p$), the number of columns in $A$ must equal the number of rows in $B$.

Determinant of a 2x2 Matrix

For $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is $\det(A) = ad - bc$.

Area Scale Factor

The absolute value of the determinant, $|\det(A)|$, representing the factor by which the matrix scales area.

Invertibility of a Matrix

A matrix is invertible if $\det(A) \neq 0$. If $\det(A) = 0$, it has no inverse.

Inverse of a Matrix

$A^{-1}$ undoes the effect of matrix $A$ during transformation, existing only when $\det(A) \neq 0$.

Identity Matrix

A matrix represented as $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ that leaves a vector unchanged.

Zero Matrix

A matrix that maps all vectors to the zero vector $\langle 0, 0 \rangle$.

Dilation Matrix

A matrix of the form $\begin{bmatrix} k & 0 \ 0 & k \end{bmatrix}$ that scales a vector by a factor $k$.

Reflection Matrix

A matrix that reflects vectors over the x-axis, represented as $\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}$.

Rotation Matrix (90° CCW)

A matrix represented as $\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}$ that rotates vectors 90 degrees counterclockwise.

Common Mistake: Matrix Multiplication Order

It is incorrect to assume that $AB = BA$ for matrices; order matters.

Common Mistake: Determinant Calculation

When calculating the inverse of a matrix, remember to divide by the determinant, $ad-bc$.

Common Mistake: Quadrant Adjustment

When finding $\theta = \tan^{-1}(b/a)$, visualize the point $(a,b)$ to adjust for the correct quadrant.