Given a projection, determine if it is linear.

Question

Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a projection onto the yz-plane. Is $T$ linear? If so, find a matrix $A$ so that $T = T_A$.

1st question I have: Does it matter that $T$ is projected on the yz plane? What would happen if it was the xy plane instead?

Anyways, I came up with the following counter example:

let $T(x) = [xy, x+y, x+z]^{\textbf{T}}$

Then $T$ is not a linear transformation because it fails the second property that defines a linear transformation $T(cX) = cT(X)$. I found that the first entry on output matrix are unequal. I got c²xy != cxy

Does my logic seem right?

Thanks.

Answer 1

If $T$ is the projection onto the $xy$-plane then $T(x,y,z)=(x,y,0)$. One can check that $T$ is linear. Applying $T$ to any basis for $\mathbb{R}^3$ yields a matrix representation for $T$