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Let $T : \mathbb{R^3} → \mathbb{R^3}$ be a linear transformation given by $T(u) = \operatorname{proj}_vU$ where $v = (2, 0,−3)$.

(a) Find the standard matrix for $T$. (b) Find a basis for the kernel of $T$.

I am completely lost on this particular question...I am familiar on finding the standard matrix and kernel but this question is a bit different...For instance, for finding Kernel of $T$, 2 vectors should be given, '$v$' and '$u$', but in this question, only '$v$' is given...

Joao
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Shawn Sagar
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    Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Joao Oct 11 '14 at 06:16

2 Answers2

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A more computationally intensive approach:

Note that the vectors $x_1=(3,0,2), x_2=(0,1,0)$ are orthogonal to $v$, hence $Tx_k = 0$. Also, $Tv = v$.

Hence in the ordered basis $v,x_1,x_2$, the operator $T$ has the form $\operatorname{diag}(1,0,0)$.

If we let $B=\begin{bmatrix} v & x_1 & x_2 \end{bmatrix}$, we have $T = B \operatorname{diag}(1,0,0) B^{-1} $.

Performing the computations gives $T = {1 \over 13}\begin{bmatrix} 4 & 0 & -6 \\ 0 & 0 & 0 \\ -6 & 0 & 9\end{bmatrix}$.

The kernel is easy to compute from the first line.

copper.hat
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Hint

$T : \mathbb{R^3} \to \mathbb{R^3}$ can be written as

$$T(x,y,z)=\frac{2x-3z}{13}(2,0,-3)=\left(\frac{4x-6z}{13},0,-\frac{6x-9z}{13}\right),$$ just making use of the definition of $\mathrm{proj}_{(2,0,-3)}(x,y,z).$ Since you say you are familiar with this, I think you can get the solution from this point.

If you get some geometric intuition of the problem it would be helpful. $T$ projects any vector on the $3$-dimensional space over a line through the origin. What happens if we project orthogonally a perpendicular vector to the line? And if the vector is parallel to the line?

mfl
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  • Still lost bro...how can I find standard matrix with only that much info...I normally have attempted questions like : Find the standard matrix of T : R3 ! R2 defined by T(x, y, z) = (3z − 2y, 4x + 11z). where two equations are given... – Shawn Sagar Oct 11 '14 at 08:01
  • @copper.hat Thank you for the comment. I have corrected the typo. – mfl Oct 11 '14 at 09:27