Linear Algebra Standard Matrix

  • #1
129
0

Homework Statement



Let [tex]\vec{u}\neq 0[/tex] be a vector in [tex]\mathbb{R}^2[/tex] and let

[tex]T:\mathbb{R}^2 \to \mathbb{R}^2[/tex] be described by

[tex]T:\vec{v} \to proj_{\vec{u}}(\vec{v})[/tex]

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

Find the standard matrix for [tex]T[/tex]

Homework Equations



[tex]proj_{\vec{u}}(\vec{v})= \frac{\vec{v}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}[/tex]

The Attempt at a Solution



Determine where [tex]T[/tex] sends [tex]\vec{e_1}[/tex] and [tex]\vec{e_2}[/tex]

[tex]f(\vec{e_1})= \frac{\vec{e_1}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}[/tex]

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

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

[tex]f(\vec{e_2})= \frac{\vec{e_2}\cdot\vec{u} }{\vec{u}\cdot\vec{u}}\vec{u}[/tex]

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

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

So does that mean that the standard matrix is

[tex]
\begin{bmatrix}
\frac{1}{2} & -\frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} \\
\end{bmatrix}
[/tex]

?

[Edited twice for LaTex mistakes]
 
Last edited:

Answers and Replies

  • #2
22,129
3,297
Seems correct!!
 
  • #3
352
0
This is correct, although your derivation in section 3 has a typo: the fourth through sixth equations should begin with [tex]f(\vec{e}_2)[/tex] rather than [tex]f(\vec{e}_1)[/tex].

It's easier to see where this operation should send the unit vectors by drawing a picture.
 
  • #4
hunt_mat
Homework Helper
1,745
26
Pretty much okay I'd say.
 
  • #5
129
0
Thank you, gentlemen. We just learned all this crazy stuff about matrix transformations and I'm still a little iffy on the concept :biggrin:
 

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