Let $T$: $R^3 \rightarrow R_3$ be the transformation $T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right)= \begin{bmatrix} 13x-2y-3z \\ 10y-2x-6z \\ 5z-3x-6y \end{bmatrix}$.
Let $H$ be the plane $x+2y+3z=0$, let $N$ be the line $N=\text{Span}\left(\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\right)$.
$(1)$ Find the standard matrix for $T$.
$(2)$ Show that the image (range) of $T$ is the plane $H$.
$(3)$ Show that $T(x)=14x$ for each vector $x$ on $H$.
$(4)$ Find all vectors $x$, for which $T(x)=0$.
$(5)$ Show that every vector $x$ can be described in an unique way as $x=n+h$, with $n$ in $N$, and $h$ $\hspace{0.6cm}$ in $H$.
$(6)$ Describe the transformation $T$ geometrically (use $5$ above).
How do I describe the transformation $T$ geometrically?
Best Answer
If you first do the five preliminary questions, I expext the geometric description of $T$ will not be so hard anymore. You can find the standard matrix representation by plugging in the standard basis vectors $(1,0,0),(0,1,0)$ and $(0,0,1)$ in $T$. The columns of the matrix are the images of these vectors. I think you'll get:
$T=\begin{pmatrix} 13&-2&-3\\ -2&10&-6\\ -3&-6&5 \end{pmatrix}$.
Furthermore, a plane in $\mathbb{R}^3$ can always be written as the set of vectors orthogonal to a normal vector of that plane. Do you know a vector orthogonal to $H$? I'm sure questions (1) - (5) will help you answer (6).