Orthonormal Sets - Find a projection matrix - Linear Algebra

[aargoo]

Homework Statement

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)

Homework Equations

proj_Sx = (x * u)u where S is a vector subspace and x is a vector

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 proj_R(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

 

[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}\}.

 

[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, proj_Ax = AA^Tx
=
[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?