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

<br /> \begin{bmatrix}<br /> \frac{1}{2} &amp; -\frac{1}{2} \\ <br /> -\frac{1}{2} &amp; \frac{1}{2} \\<br /> \end{bmatrix}<br />

?

[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