Orthogonal projection on Span

Question

For the vectors: $$v_1 = \frac {1}{2} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix},\ v_2 =\frac {1}{\sqrt2} \begin{pmatrix} -1\\ 1 \\ 0 \\ 0 \end{pmatrix}$$

Find an orthogonal projection for $(-1,1,2,2)^T$ onto Span$(v_1,v_2)$, which is a subspace of $V$.

I am unsure how to find the orthogonal projection when I have to do it onto the Span of vectors, I hope somebody can show a method to calculate it or give a hint.

Answer 1

HINT

• consider the matrix $A=[v_1\quad v_2]$
• the projection matrix is $P=A(A^TA)^{-1}A^T$

Answer 2

The spanning set is orthonormal, so you simply add up the projections onto them: $$(v\cdot v_1)v_1+(v\cdot v_2)v_2.$$ As saulspatz hinted, this is exactly what you do to find the coordinates of $v$ relative to the standard basis, i.e., $v=x_v\mathbf e_1+y_v\mathbf e_2=(v\cdot\mathbf e_1)\mathbf e_1+(v\cdot\mathbf e_2)\mathbf e_2$.