###### Example 2.2.1

Suppose we have the matrix \(A\) and vector \(\xvec\) as given below.

Their product will be defined to be the linear combination of the columns of \(A\) using the components of \(\xvec\) as weights. This means that

Let's take note of the dimensions of the matrix and vectors. The two components of the vector \(\xvec\) are weights used to form a linear combination of the columns of \(A\text{.}\) Since \(\xvec\) has two components, \(A\) must have two columns. In other words, the number of columns of \(A\) must equal the dimension of the vector \(\xvec\text{.}\)

In the same way, the columns of \(A\) are 3-dimensional so any linear combination of them is 3-dimensional as well. Therefore, \(A\xvec\) will be 3-dimensional.

We then see that if \(A\) is a \(3\times2\) matrix, \(\xvec\) must be a 2-dimensional vector and \(A\xvec\) will be 3-dimensional.