let $L$ be the line in $\mathbb{R}^3$ space determined by the vector (1,1,1)T. Let $P$ be the projection matrix for the projection onto the line $L$. Determine the matrix $P$ and then:
a) Determine the orthonormal basis of the orthonormal complement $L \perp $ of the line $L$.
b) Determine the projection of the vector (1,2,3)T on the line $L$.
c) Determine the projection of the vector (1,2,3)T onto the orthogonal complement $L \perp$ of the line L.
Could you help me to understand and solve this task? How to find matrix $P$, orthogonal complement, projection of vector on plane...?
(a) Use Gram-Schmidt on e.g. the matrix $\begin{pmatrix}1&0&0\\1&1&0\\1&0&1\end{pmatrix}.$ You will get as first column vector a normalized vector pointing in direction of $L$, but the other two vectors will be an orthonormal basid of $L^\bot$
For (b) use wikipedia formulas with respect to the orthonormal basis found in 1.
For (c) dito (b) but with the complement basis.