A fundamental idea in linear algebra is to view a linear combination of vectors as the product of a matrix and a vector

Existence of Solution of Ax = B

The equation Ax = b has a solution if and only if b is a linear combination of the columns of

which is to say -

• is b in span{a1,.. an}
• is Ax = b consistent

Example

Quick Note

When can we say that Ax = b is true for all possible b ?