The Matrix Equation 1.4

What does a matrix equation represent?

A matrix equation represents a linear combination of vectors as a matrix times a vector

Write out system of linear equations as a vector equation

Write out vector equation as a matrix equation

The general matrix equation has the form

Since the system of equations, the vector equation, and the matrix equation are all equivalent…

They all have the same solution

1. b is a linear combination of vectors V1, V2, …, Vn

2. b is in the span of V1, V2,…, Vn

3. System of equations, the vector equation, and the matrix equation are consistent

Let A be an Mxn matrix, then the statements are equivalent

1. AX = b has a solution for all b in R^m

2. Each b is a linear combination of the columns of A

3. The columns of A span R^m

4. A has a pivot position in every row