Find the orthogonal projection of vector $b$ onto column space of given matrix $A$, then find orthogonal projection of the same vector onto kernel of matrix $A^T$.
$$A=\begin{bmatrix} 0 & 1 \\ 1 & 3 \\ -1 & 1 \\ 0 & 0 \\ -1 & 1 \end{bmatrix}$$
$$b=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$
Ok i know what is column space, it is simply image of $A$ wich is obviosely span of columns of $A$ since they are linearly independent, now, when it comes to orthogonal projections, all i know is that $proj_\vec{a}\vec{b}=\frac{\vec{a}\vec{b}}{||\vec{a}||}\vec{a}$. Is this the same formula that i can use here? Because even if it is, how could i calculate dot product of $b$ and column space?
Best Answer
The formula you mentioned is about projections on vectors. The problem here is about projections on spaces.
Determine an orthogonal basis $\{e_1,e_2\}$ of the space spanned by the collumns, using Gram-Schmidt. Then the projection of $b$ is $\langle b,e_1\rangle e_1+\langle b,e_2\rangle e_2$.