Let $T: R_3 \to R_3$ be the transformation that reflects vector $x=(x_1, x_2, x_3)$ through the plane $x_3=0$ onto $T(x) = (x_1, x_2, -x_3)$.

Question

Let $T: R_3 \to R_3$ be the transformation that reflects vector $x=(x_1, x_2, x_3)$ through the plane $x_3=0$ onto $T(x) = (x_1, x_2, -x_3).$

I am trying to find if $T$ is a linear transformation and if not why.

I have no idea how to start this problem. Can anyone give me some kind of direction and help me figure out this problem step by step? I have been working on figuring out how to answer linear transformation problems, but the way this one is worded just throws me off. Thanks!

So far I have answered:

Let $R_2$ to $R_3$ be the transformation defined by $T(x_1, x_2) = (2x_1-3x_2, x_1+4, 5x_2)$ I calculated that the above was not a linear transformation because $T(u+v) = T(u) + T(v)$ was not satisfied. Hopefully I am on the right track here with that conclusion. So help me out!!

Answer 1

I'll begin the proof for you:

Let $a = (a_1, a_2, a_3), b=(b_1, b_2, b_3)\in\Bbb R^3$ and $\lambda \in\Bbb R$. Then

$$\begin{align}T[a+\lambda b] &= T(a_1+\lambda b_1, a_2+\lambda b_2, a_3+\lambda b_3) \\ &= (a_1+\lambda b_1, a_2+\lambda b_2, -(a_3+\lambda b_3)) \\ &\qquad\vdots \\ &\stackrel{?}=T(a)+\lambda T(b)\end{align}$$

See if you can get to that last step or not and you'll be done.

Answer 2

Let $x = (x_1,x_2,x_3)$ and $y = (y_1,y_2,y_3)$ and $p $ in $ R$, then $T(x+p*y) = (x_1+p*y_1,x_2+p*y_2,-(x_3+p*y_3)) = (x_1+p*y_1,x_2+p*y_2,-x_3 -p*y_3) = (x_1,x_2,-x_3) + (p*y_1,p*y_2,-p*y_3) = T(x) + p*T(y)$

Answer 3

You just need to verify the condition $T(aX+by)=aT(X)+bT(Y)$ where $X,Y\in\mathbb R^3$ and $a,b$ are scalars in $\mathbb R$.