3.3 Matricies & Vectors


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3.3 Contents


3.3.1 Matrix Definition & Notation

A representation of a linear transformations with components as constant multipliers. It can be used to represent a system of lines, points, or angles of rotation. $$ A= \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} $$ The elements by themselves are represented with $A_{ij}$

3.3.2 Identity Matrix

The matrix form of the multiplicative identity. The diagonal from top left to bottom right is called the (as in only) diagonal, and its sum is called a trace. $$ M·M^{-1}=I= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
Jump to Matrix Arithmetic
Jump to Determinant

3.3.3 Matrix Arithmetic

Scalar Matrix
A number $n$ by itself is called a scalar value, and can be converted to a square matrix with the identity matrix $n·I$ $$ n→ \begin{bmatrix} n & 0 \\ 0 & n \end{bmatrix} $$
Scalar Distribution
Scalar coefficients distribute throughout $$ n·A= \begin{bmatrix} n·A_{11} & n·A_{12} \\ n·A_{21} & n·A_{22} \end{bmatrix} $$
Addition & Subtraction
Matrices may be added only if they have the same dimensions. Matrix addition and subtraction commutes and associates. $$ A\pm B= \begin{bmatrix} A_{11}\pm B_{11} & A_{12}\pm B_{12} \\ A_{21}\pm B_{21} & A_{22}\pm B_{22} \end{bmatrix} $$
Product
Matrices may be multiplied regardless of dimensions (as in the starting pattern below), with the dimensions of the product being $A_{ij}·B_{kl}=C_{il}$. Matrix multiplication does not commute. $$A·B=\begin{bmatrix} A_{11}·B_{11}+A_{12}·B_{21} & A_{11}·B_{12}+A_{12}·B_{22} \\ A_{21}·B_{11}+A_{22}·B_{21} & A_{21}·B_{12}+A_{22}·B_{22} \end{bmatrix} $$

3.3.4 Determinant

The change in area when an identity matrix undergoes linear transformation to arrive at a matrix $$ \text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} =a·d-b·c $$
Image taken from Matthew N. Bernstein and edited

3.3.5 Matrix Solutions with Determinants (Cramer's Rule)

Given a system of equations $$ \begin{bmatrix} \begin{array}{cc|c} x_1 & y_1 & c_1 \\ x_2 & y_2 & c_2 \end{array} \end{bmatrix} $$ Intersections can be solved using the determinant with determinants where the solutions column becomes the transformation for that variable $$ D\phantom{x}= \begin{vmatrix} \begin{array}{cc} x_1 & y_1 \\ x_2 & y_2 \end{array} \end{vmatrix} \qquad \phantom{x=\frac{Dx}{D}} $$ $$ Dx= \begin{vmatrix} \begin{array}{cc} c_1 & y_1 \\ c_2 & y_2 \end{array} \end{vmatrix} \qquad x=\frac{Dx}{D} $$ $$ Dy= \begin{vmatrix} \begin{array}{cc} x_1 & c_1 \\ x_2 & c_2 \end{array} \end{vmatrix} \qquad y=\frac{Dy}{D} $$

3.3.6 Inverse Matrix

$$ A^{-1}= \begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \end{bmatrix}^{-1} = \frac{1}{\text{det}(A)}· \begin{bmatrix} y_2 & -y_1 \\ -x_2 & x_1 \end{bmatrix} $$
Properties
Proof by Standard Inversion
Join $A$ with $I$ $$ \begin{bmatrix} \begin{array}{cc|cc} x_1 & y_1 & 1 & 0 \\ x_2 & y_2 & 0 & 1 \end{array} \end{bmatrix} $$ Multiply row 1 by $1/x_1$ $$ \begin{bmatrix} \begin{array}{cc|cc} 1 & y_1/x_1 & 1/x_1 & 0 \\ x_2 & y_2 & 0 & 1 \end{array} \end{bmatrix} $$ Subtract row 1 multiplied by $x_2$ from row 2 $$ \begin{bmatrix} \begin{array}{cc|cc} 1 & y_1/x_1 & 1/x_1 & 0 \\ 0 & y_2-x_2·y_1/x_1 & -x_2/x_1 & 1 \end{array} \end{bmatrix} $$ Factor and rewrite $A_{22}$ $$y_2-\frac{x_2·y_1}{x_1}=\frac{x_1·y_2-x_2·y_1}{x_1}→\frac{\text{det}(A)}{x_1}$$ Multiply row 2 by $x_1/\text{det}(A)$ $$ \begin{bmatrix} \begin{array}{cc|cc} 1 & y_1/x_1 & 1/x_1 & 0 \\ 0 & 1 & -x_2/\text{det}(A) & x_1/\text{det}(A) \end{array} \end{bmatrix} $$ Subtract row 2 multiplied by $y_1/x_1$ from row 1 $$ \begin{bmatrix} \begin{array}{cc|cc} 1 & 0 & 1/x_1+x_2·y_1/\big(x_1·\text{det}(A)\big) & -y_1/\text{det}(A) \\ 0 & 1 & -x_2/\text{det}(A) & x_1/\text{det}(A) \end{array} \end{bmatrix} $$ Factor what is now $A^{-1}_{11}$ $$\frac{1}{x_1}+\frac{x_2·y_1}{x_1·\text{det}(A)}=$$ $$\frac{\text{det}(A)}{x_1·\text{det}(A)}+\frac{x_2·y_1}{x_1·\text{det}(A)}=$$ $$\frac{x_1·y_2-x_2·y_1+x_2·y_1}{x_1·\text{det}(A)}=\frac{y_2}{\text{det}(A)}$$ Unjoin $I$ from $A^{-1}$, and factor out $\text{det}(A)$

3.3.7 Transpose Matrix

A matrix that's been flipped about its trace $$ M= \begin{bmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{bmatrix} \qquad M^T= \begin{bmatrix} M_{11} & M_{21} \\ M_{12} & M_{22} \end{bmatrix} $$
Properties

3.3.8 Vector Definition & Notation

A number with a direction and magnitude $$\vec{v}=v_x\hat{\text{i}}+v_y\hat{\text{j}}$$
Jump to Complex Unit Circle
Properties

3.3.9 Vector Magnitude & Unit Vector

The distance between two points $P,Q$ written as $\overrightarrow{PQ}=\lang x_2-x_1,y_2-y_1\rang$ is $$\big|\big|\overrightarrow{PQ}\big|\big|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ It follows that if a vector is positioned at the origin, then $$\|\overrightarrow{v}\|=\sqrt{v_x^2+v_y^2}$$ A vector $\vec{u}$ at the origin with a length of 1 parallel to vector $\vec{v}$ is its unit vector $$\pm\vec{u}=\pm\frac{\vec{v}}{\|\vec{v}\|}$$

3.3.A Vector Arithmetic

All fundamental properties and distribution apply to vectors
Addition & Subtraction
Images taken from Varsity Tutors and edited
$$ \vec{u} \pm \vec{v} = \lang \vec{u}_1 \pm \vec{u}_2 , \vec{v}_1 \pm \vec{v}_2 \rang $$
Distributive Property
$$c·\vec{v}=\lang c·v_1,c·v_2\rang$$

3.3.B Dot Product 🔧

Dot Product Using Components
Coordinate Unit Vectors
$$\hat{\text{i}}·\hat{\text{i}}=\hat{\text{j}}·\hat{\text{j}}=1$$
Geometry
$$\vec{\text{u}}\parallel\vec{\text{v}}\medspace\text{ if }\medspace\vec{\text{u}}·\vec{\text{v}}=\pm\|\vec{\text{u}}\|·\|\vec{\text{v}}\|$$ $$\vec{\text{u}}\perp\vec{\text{v}}\medspace\text{ if }\medspace\vec{\text{u}}·\vec{\text{v}}= $$

3.3.C Orthogonal Projection 🔧


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