Matrix of orthogonalprojection

Question

whats the matrix of an orthogonalprojection?

This question was part of a longer task:

1. Step: two vectors were given:
   • $v_1 = (i, 0, 1)$
   • $v_2 = (0,i,1)$

U is a vector space, spanned by v1 and v2

2. Step: Determine a orthonormalbasis (via gram-schmid) of U

3. Step: complement this basis to an orthonormalbasis of $\mathbb{C}3$.

4. Step: What is the coordinate-matrix (relating to the basis of Step 3) for the orthogonalprojection onto U?

??

Answer 1

Orthogonal projection is in this such a linear thansformation, that if you have basis $\{v_1,v_2\}$ of $U$ and orthogonal basis $\{v_1,v_2,v_3\}$ then for all $x=av_1+bv_2+cv_3 \in \mathbb{C}^3$:

$$P(x)=av_1+bv_2$$

So $PV_1=v_1$, $Pv_2=v_2$ and $Pv_3=0$ and finally:

$$P=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}$$