# Prove whether the following transformation is linear or not.

linear-transformationsmatrices

I'm not sure if I'm going wrong with the proof, but I'm getting my answer as, the transformation is linear, whereas according to our teacher, it is not supposed to be linear.

Q. $$T(v_1, v_2) = (v_1, v_1)$$ is not a linear transformation. Prove it.

My Working:
So, since this is in two dimensional space, as the input and output are 2D coordinates, or matrices, the same two rules for linear transforms apply, except the inputs are matrices
$$T(a_1+ a_2)= T(a_1)+ T(a_2)$$
$$T(c*a_1) = c*T(a_1)$$
where $$a_1$$ and $$a_2$$ are $$2\times 1$$ matrices.

Now, let $$a_1 = \begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix}$$
let $$a_2 = \begin{bmatrix} w_1 \\ w_2 \\ \end{bmatrix}$$
$$T(a_1+ a_2)= T(a_1)+ T(a_2)$$
$$T(a_1+ a_2)= T\begin{bmatrix} v_1 \\ v_2 \\ \end{bmatrix}+ T\begin{bmatrix} w_1 \\ w_2 \\ \end{bmatrix}=\begin{bmatrix} v_1 \\ v_1 \\ \end{bmatrix}+ \begin{bmatrix} w_1 \\ w_1 \\ \end{bmatrix}= \begin{bmatrix} v_1+w_1 \\ v_1 +w_1 \\ \end{bmatrix}$$

which is true.

The same obviously holds true for the second statement as well.

Thus, I am able to prove that since the linear transformation satisfies the two equations, the transformation is linear. I'm not sure if I've gone wrong somewhere, in which case I'd really appreciate any help!

You are right. You can remind your teacher that every matrix generates some linear transformation. In our case it is $$A=\begin{pmatrix} 1 &0 \\ 1 &0 \end{pmatrix}.$$ Certainly $$A\cdot\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} v_1 \\ v_1 \end{pmatrix}.$$ I am assuming that $$(v_1,v_2)$$ are the coordinates of a vector in two-dimensional space.