so assume we have a linear transformation. L which takes

For the matrix A, you can write the 3rd column as the linear combination of the first and second columns.

since a linear transformation has the property L( )= then

if the first column of A went to the first column of B, and the second column of A went to the second column of B then

since this is not equal to the 3rd column of B, there is no Linear transformation which takes the first column of A to the first column of B, second ....