Let T R2 rightarrow R2 be the linear transformation defined

Let T : R^2 rightarrow R^2 be the linear transformation defined by T(x, y) = (- 12 x - 4y, 9x + 3y) Find a vector w that is not in the image of T.

Solution

T(1,0)=(-12,9)

T(0,1)=(-4,3)

These are images of the two standard basis vectors of R2.

Note that : T(1,0)=3T(0,1)

So all vectors in the image will be of the form:

aT(0,1)=a(-4,3)=(-4a,3a)

So one vector not in image T would be:

(4,3)