# Orthonormal Sets - Find a projection matrix - Linear Algebra

1. Jul 21, 2014

### aargoo

1. The problem statement, all variables and given/known data
Let A be the 4x2 matrix
|1/2 -1/2|
|1/2 -1/2|
|1/2 1/2|
|1/2 1/2|

Find the projection matrix P that projects vectors in R4 onto R(A)

2. Relevant equations
projSx = (x * u)u where S is a vector subspace and x is a vector

3. The attempt at a solution
v1 = (1/2, 1/2, 1/2, 1/2)T
v2 = (-1/2, -1/2, 1/2, 1/2)T
v1v2 = 0, hence the vectors are orthogonal
||v1|| = 1
||v2|| = 1, hence they form an orthonormal basis for R2

R(A) = span{(1/2, -1/2)T, (1/2, -1/2)T, (1/2, 1/2)T, (1/2, 1/2)T} = span{(1/2, -1/2)T, (1/2, 1/2)T}

And from here I am a bit lost. Would I define x as the standard basis for R4 and find projR(A)x ?

The answer from the book is given as
[.5 -.5 0 0]
[-.5 .5 0 0]
[0 0 .5 -.5]
[0 0 -.5 .5]

Thanks

Note: This is from a section before the Gram-Schmidt orthogonalization process

Last edited: Jul 21, 2014
2. Jul 21, 2014

### HallsofIvy

The first thing I would do is determine R(A). Given any vector v= [x, y], we have
$$Av= \begin{bmatrix}\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}\frac{1}{2}x- \frac{1}{2}y \\ \frac{1}{2}x- \frac{1}{2}y \\ \frac{1}{2}x+ \frac{1}{2}y \\ \frac{1}{2}x+ \frac{1}{2}y\end{bmatrix}$$.

We can write that result as
$$\begin{bmatrix}a \\ a \\ b \\ b\end{bmatrix}= \begin{bmatrix}a \\ a \\ 0 \\ 0 \end{bmatrix}+ \begin{bmatrix}0 \\ 0 \\ b \\ b \end{bmatrix}= a\begin{bmatrix}1 \\ 1 \\ 0 \\ 0 \end{bmatrix}+ b\begin{bmatrix}0 \\ 0 \\ 1 \\ 1 \end{bmatrix}$$
where a= x/2+ y/2 and b= x/2- y/2. That is, a basis for R(A) is $$\{\begin{bmatrix}1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1\end{bmatrix}\}$$.

3. Jul 22, 2014

### aargoo

Hmm, I'm still trying to figure out if I'm missing a formula somewhere...
A =
[1 0
1 0
0 1
0 1]

So, projAx = AATx
=
[1 1 0 0
1 1 0 0
0 0 1 1
0 0 1 1] x

I'm now confused as to how to find x? It should be a 4x4, correct?