Matrix Vector Product

matrix_multiplication

Symbol	Meaning
$\in$	belongs to
${\mathbb{R}}^n$	the set of vectors with n real-valued elements
${\mathbb{R}}^{m\times n}$	the set of real-valued matrices with m rows and n columns

Definition of Matrix Vector Multiplication

If $A\in {\mathbb{R}}^{m\times n}$ has vectors ${\mathbf{a}}_1,\dots ,{\mathbf{a}}_{\mathbf{n}}$ and $\mathbf{x}\in {\mathbb{R}}^n,$ then the matrix Vector Product $A\mathbf{x}$ is a linear combination of the columns of $A$. It can be shown as shown: A \mathbf{x} = \left[ \matrix{| & | & \dots & | \\ \mathbf{a_1} & \mathbf{a_2} & \dots & \mathbf{a_n} \\ | & | & \dots & |} \right] \left[ \matrix {x_1 \\ x_2 \\ \vdots \\ x_n} \right] = x_1\mathbf{a_1} + x_2\mathbf{a_2} + \dots +x_n\mathbf{a_n} = \left[ \matrix{ \vdots \\ x_1\mathbf{a_1} + x_2\mathbf{a_2} + \dots +x_n\mathbf{a_n} \\ \vdots } \right] NOTE: $A\mathbf{x}$ is always in the span of $A$

Existence of Solutions

The equation $A\mathbf{x}=\mathbf{b}$ has a solution if and only if $\mathbf{b}$ is a linear combination of the columns of $A$.

Example

For what vectors $$\begin{gathered} \mathbf{b}=\left[\text{matrix}{b}_1 \\ {b}_2 \\ {b}_3\right] \end{gathered}$$ does the equation have a solution?

$\left[\begin{array}{cccc}1& 3& 4& b_1\\ 2& 8& 4& b_2\\ 0& 1& -2& b_3\end{array}\right]$ $\left[\begin{array}{cccc}1& 3& 4& b_1\\ 0& 2& -4& b_2-2b_1\\ 0& 0& 0& b_3-\frac{1}{2}{b}_2+b_1\end{array}\right]$ Solving for $b_1$ from the last row, $b_1=\frac{1}{2}-b_3$ $$\begin{gathered} \boxed{\mathbf{b}=\left[\text{array}\frac{1}{2}-b_3 \\ {b}_2 \\ {b}_3\right]} \end{gathered}$$