Matrix-Vector Product

Definition

Let , and . Then

Example

Consider this linear system of equations

Let

So that . The system is consistent if and only if we can find so that:

That is, the system is consistent if and only if can be expressed as a linear combination of the columns of .

With our definition, we can write this as , or

Observation

A system of linear equations solved by a matrix is actually the matrix-vector product

Recall that with , is the constant matrix, and is the coefficient matrix.

Theorem (Kinda useless)

Every linear system of equations can be expressed as for some matrix and some vector
The system is consistent if and only if can be expressed as a linear combination of the columns of (i.e each can be found)
If are the columns of and , then satisfies if and only if (i.e the values of must solve the linear system)

Important

Note the sizes of the vectors and matrices:

Example

The matrix-vector product does not exist because has two columns but has 3 components.

Properties

With Dot Product

Definition

Given , there are vectors such that:

And for any :

Example

Given
We have

If we define:

Definition

Denoted by or or , is the square matrix with size with for .

Examples

The Interesting Property

Theorem

Let . If for all , then