Show that the linear transformation T is not surjective by finding an element of the codomain, v, such that there is no vector u with T (u) = v

We wish to find an output vector v that has no associated input. This is the same as requiring that there is no solution to the equality

In other words, we would like to find an element of C3 not in the set

If we make these vectors the rows of a matrix, and row-reduce, Theorem BRS provides an alternate description of Y .

If we add these vectors together, and then change the third component of the result, we will create a vector  that lies outside of Y ,