# Computing the standard matrix of the linear transformation

Can you please explain this question to me?

Suppose that $w = [1,2,3]^T$ and $L: \mathbb{R}^3\to \mathbb{R}^3$ is defined by $L(x) =\text{Proj}_w(x)$ (projection of $x$ onto $w$). Compute the standard matrix of $[L]$ of the linear transformation.

Sorry for the format, i don't know how to use the website very well

Hint: L(x) is a linear transformation

I couldn't compute the projection because $x$ is missing. Thanks.

• Is $L$ an orthogonal projection? Otherwise, $L$ is a projection parallel to what onto $w$? – C. Falcon Jun 19 '16 at 20:47
• Is it a linear transformation? – Majid Jun 19 '16 at 20:47
• sorry i forgot to write the hint. L is a linear transformation – matheu96 Jun 19 '16 at 20:49

The matrix that projects onto the $1$-dimensional vector space spanned by $\mathrm{w}$, which is a line, is

$$\mathrm{P}_{\mathrm{w}} := \dfrac{\mathrm{w} \mathrm{w}^T}{\mathrm{w}^T \mathrm{w}} = \frac{1}{14} \begin{bmatrix} 1 & 2 & 3\\ 2 & 4 & 6\\ 3 & 6 & 9\end{bmatrix}$$

• thanks, can you please explain it to me – matheu96 Jun 19 '16 at 21:33
• @matheu96 Let $\mathrm{x}$ be a scalar multiple of $\mathrm{w}$. Compute $\mathrm{P}_{\mathrm{w}} \mathrm{x}$. Then compute $\mathrm{P}_{\mathrm{w}}^2$. – Rodrigo de Azevedo Jun 19 '16 at 21:36
• will the final answer be the result of (P_w)^2 – matheu96 Jun 19 '16 at 23:21
• @matheu96 If $P$ is a projection matrix, then $P^2 = P$. – Rodrigo de Azevedo Jun 19 '16 at 23:25