Section 1 definitions

Linearly Independent

$\vec{v_1},\vec{v_2}, ... , \vec{v_n}$ & $c_1\vec{v_1}, c_2\vec{v_2}, ... , c_n\vec{v_n} = \vec{0}$ is only true when $c_1, c_2, c_3, ... , c_n = 0$

Matrix Addition

(1)
\begin{align} \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 2 & 4\\5 & 3 \end{bmatrix} = \begin{bmatrix} 1+2 & 2+4\\3+5 & 3+4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 8 & 7 \end{bmatrix}. \end{align}

Multiplication

How a vector acts on something.

Plane%20Diagram.png

If A is an m x n matrix and B is an n x p matrix where B = $\vec{b_1}, \vec{b_2}, \vec{b_3}, ... , \vec{b_p}$ then AB is $A\begin{bmatrix}\vec{b_1}&\vec{b_2}&\vec{b_3}&...&\vec{b_n} \end{bmatrix}$

A = $\begin{bmatrix} 1 & 2 & 3\\0&1&0 \end{bmatrix}$, B = $\begin{bmatrix} 1&0&1\\0&1&0\\1&0&1 \end{bmatrix}$

AB = $\begin{bmatrix} 1(1)+2(0)+3(1)&1(0)+2(1)+3(0)&1(1)+2(0)+3(1)\\0(1)+1(0)+0(1)&0(0)+1(1)+0(0)&0(1)+1(0)+0(1) \end{bmatrix} = \begin{bmatrix} 4&2&4\\0&1&0 \end{bmatrix}$

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