# Homework Help: Projection onto subspace along subspace

1. Apr 14, 2009

### descendency

1. The problem statement, all variables and given/known data
Find a projection [matrix] E which projects R2 onto the subspace spanned by (1,-1) along the subspace spanned by (1,2).

2. Relevant equations
$$P = \frac{a a^{T}}{a^{T} a}$$

3. 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?

Last edited: Apr 15, 2009