Projection onto subspace along subspace

[descendency]

Homework Statement

Find a projection [matrix] E which projects R² onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).

Homework Equations

$$P = \frac{a a^{T}}{a^{T} a}$$

The Attempt at a Solution

Computing P...
$$P = \[ \left( \begin{array}{ccc} \frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{2} & \frac{1}{2} \end{array} \right)\]$$

Let D be a change of basis matrix from the standard basis to the basis B = {(1,-1), (1,2)}
$$D = \[ \left( \begin{array}{ccc} 1 & 1 \\ -1 & 2 \end{array} \right)\]$$

E = D^-1PD?

E² = D^-1PDD^-1PD = D^-1P²D = D^-1PD, so it passes that test for being a projection.

Is E the projection talked about in the question?

 

[mighty2000]

Yes, E is the projection matrix that projects R2 onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).