Using Matrices to Find Basis of Image of Linear Transformation

Hi all ---

I'm having trouble understanding one part of this example from my textbook. It's using the matrix of a linear transformation to find the basis of the image of the same linear transformation.

**Question ---**

Let be a linear transformation defined by:

.

Let be a basis for and

let be a basis for .

If , use it to find the basis for the image of .

**Textbook Solution ---**

Recall that is in .

But is already in row-reduced format,

so

This is what I don't understand. How do you know that the vectors in the here are coordinate vectors? An obvious answer - but not the one I'm looking for - is because these vectors aren't in or . But what's the true math answer? Thanks a lot ---

so .

Re: Using Matrices to Find Basis of Image of Linear Transformation

__Hint__: The linear transformation can be written in the way:

http://quicklatex.com/cache3/ql_369e...77389c5_l3.png

where the column vectors span and are expressed in coordinates on .