Project R4x1 onto R along N: Find Rule

[mddoyle]

Homework Statement

Let R={[x,y,z,w]:x=y and z=w} and N={[x,y,z,w]:x=-y and z=-w}
Find the rule in standard coordinates for the projection of R4x1 onto R along N

Homework Equations

The Attempt at a Solution

I have B wrt R as {[1,1,0,0],[0,0,1,1]} and B wrt N as {[1,-1,0,0], [0,0,1,-1]}, so my basis is {[1,1,0,0],[0,0,1,1],[1,-1,0,0],[0,0,1,-1]}.
[E]B=
[1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0]

I used my basis as my Q^(-1), so
Q=
[.5 .5 0 0
0 0 .5 .5
.5 -.5 0 0
0 0 .5 -.5]
which gives me
Q^(-1)[E]BQ=
[.5 .5 0 0
.5 .5 0 0
0 0 .5 .5
0 0 .5 .5]
so my rule would be
E[x =
y
z
w]

[x+y
x+y
z+w
z+w]

but,
E[1 =
1
0
0]

[2
2
0
0]

instead of
[1
1
0
0]

but
E[1 =
-1
0
0]

[0
0
0
0]
which is correct

my Q^(-1) can't be right, but I'm not sure how to find another one.

 

[HallsofIvy]

Sorry, but I find this impossible to read!