Construct an Orthogonal basis for the subspace $W\subseteq R^3$. given by the following basis:
$\{\begin{pmatrix} 3 \\ 0 \\ -1 \\ \end{pmatrix} \}$, $\{\begin{pmatrix} 8 \\ 5 \\ -6 \\ \end{pmatrix} \}$
Okay so I have never been asked to find a basis given a basis set. This is how I did it, is this correct?
I made the first vector in the basis x and the second y. so
X= $\{\begin{pmatrix} 3 \\ 0 \\ -1 \\ \end{pmatrix} \}$ and y=$\{\begin{pmatrix} 8 \\ 5 \\ -6 \\ \end{pmatrix} \}$
Then using the formulas: q1= x/$\|x\|$, y'=y-($q1^T$*y)*q1 and y'/$\|y'\|$.
I get the following basis:
$\frac{1}{\sqrt{10}}$$\{\begin{pmatrix} 3\\ 0 \\ -1 \\ \end{pmatrix} \}$ and $\frac{1}{\sqrt{35}}$$\{\begin{pmatrix} -1 \\ 5 \\ -3 \\ \end{pmatrix} \}$