Matrix Vector Product

The product of a matrix and a vector is as shown:

\[[\vec a_1, \vec a_2, \cdots, \vec a_n]\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = x_1\vec a_1 + x_2\vec a_2 + \cdots + x_n\vec a_n\]

A matrix vector product is well-defined if the number of columns in the matrix equals the number of rows in the vector. The output is a vector with the length of the number of rows in the matrix.

Matrix Equation

\[A\vec x = \vec b\]

If a system has two distinct solutions, it must have infinitely many solutions.

Properties

1. $A(\vec x+\vec y)=A\vec x + A\vec y$
2. $A(c\vec x)=c(A\vec x)$