Linear Algebra Standard Matrix

[Screwdriver]

Homework Statement

Let $$\vec{u}\neq 0$$ be a vector in $$\mathbb{R}^2$$ and let

$$T:\mathbb{R}^2 \to \mathbb{R}^2$$ be described by

$$T:\vec{v} \to proj_{\vec{u}}(\vec{v})$$

If $$\vec{u}=[1,-1]$$

Find the standard matrix for $$T$$

Homework Equations

$$proj_{\vec{u}}(\vec{v})= \frac{\vec{v}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}$$

The Attempt at a Solution

Determine where $$T$$ sends $$\vec{e_1}$$ and $$\vec{e_2}$$

$$f(\vec{e_1})= \frac{\vec{e_1}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}$$

$$f(\vec{e_1})= \frac{[1,0]\cdot [1,-1]}{[1,-1]\cdot [1,-1]}$$

$$f(\vec{e_1})= [\frac{1}{2},-\frac{1}{2}]$$

$$f(\vec{e_2})= \frac{\vec{e_2}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}$$

$$f(\vec{e_2})= \frac{[0,1]\cdot [1,-1]}{[1,-1]\cdot [1,-1]}$$

$$f(\vec{e_2})= [-\frac{1}{2},\frac{1}{2}]$$

So does that mean that the standard matrix is

$$\begin{bmatrix} \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} \\ \end{bmatrix}$$

?

[Edited twice for LaTex mistakes]

 

[micromass]

Seems correct!

 

[ystael]

This is correct, although your derivation in section 3 has a typo: the fourth through sixth equations should begin with $$f(\vec{e}_2)$$ rather than $$f(\vec{e}_1)$$.

It's easier to see where this operation should send the unit vectors by drawing a picture.

 

[hunt_mat]

Pretty much okay I'd say.

 

[Screwdriver]

Thank you, gentlemen. We just learned all this crazy stuff about matrix transformations and I'm still a little iffy on the concept