Is the following a linear transformation?

Question

Is the following a linear transformation? Show that it preserves addition and scalar multiplication. $$T:\Bbb R^3 \to \Bbb R^3 \text{ defined by } T(\vec x) = \begin{bmatrix} \vec x * \vec x\\0\\0\end{bmatrix}.$$ Note that $\vec x * \vec x$ is the dot product.

I understand that in order for this to be a linear transformation, it needs to preserve addition, $T(\vec x + \vec y)=T(\vec x)+T(\vec y)$ and scalar multiplication, $T(k \vec x) = kT(\vec x)$.

I think I am confused because of the dot product within the transformation. Can anyone help me start this problem? Thanks!

Answer 1

Since $$T(x+y)=\begin{pmatrix}\langle x+y,x+y\rangle \\ 0\\ 0\end{pmatrix}=\begin{pmatrix}\langle x,x\rangle \\ 0\\ 0\end{pmatrix}+\begin{pmatrix}2\langle x,y\rangle \\ 0\\ 0\end{pmatrix}+\begin{pmatrix}\langle y,y\rangle \\ 0\\ 0\end{pmatrix}\neq T(x)+T(y)$$ for $x,y\neq 0$, the operator $T$ is not linear.

Answer 2

I don't think it's linear. For instance, since $x\cdot x=\mid x\mid^2$, if we multiply $x$ by $k\neq1,0$, we get $T(kx)=k^2T(x)\neq kT(x)$.