Matrix-Vector Product

Introduced to in Math 115

[! Definition 3.2.2 Matrix-Vector Product]

Let A = [${\vec{a}}_1\times {\vec{a}}_n$] $\in M_{m\times n}(\mathbb{R})$ and $\vec{x}$ = $\left[\begin{array}{c}{x}_1\\ \dots \\ x_n\end{array}\right]\in {\mathbb{R}}^n$. Then the vector A$\vec{x}$ is defined

A$\vec{x}$ = $x_1{\vec{a}}_1+\cdots +x_n{\vec{a}}_n\in {\mathbb{R}}^m$

[! Matrix Equality Theorem]

Let A, B $\in M_{m\times n}(\mathbb{R})$. If A$\vec{x}$ = B$\vec{x}$ for every $\vec{x}\in {\mathbb{R}}^n$, then A = B

[! Properties of Matrix Vector Product]

Let A, B $\in M_{m\times n}(\mathbb{R}),\vec{x},\vec{y}\in {\mathbb{R}}^n$ and c $\in \mathbb{R}$ Then

a) A($\vec{x}+\vec{y}$) = $A\vec{x}+A\vec{y}$

b) A(c$\vec{x}$) = c(A$\vec{x}$) = (cA)$\vec{x}$

c) (A + B)$\vec{x}$ = A$\vec{x}+B\vec{x}$

The n x n identity matrix, denoted by I${}_n$ (or I${}_{n\times n}$ or just I if the size is clear) is the square matrix of size n $\times$ n with (I${}_n$)${}_{ii}$ = 1 for i = 1, 2, $\dots$, n and zeros elsewhere