I have a question that says:
Use the Gram-Schmidt orthonormalization process to transform the given basis for a subspace into an orthonormal basis for the subspace. Be sure to show your matrix $B'$ and then your matrix $B''$. $B = > {(1,2,0), (2,0,-2)}$
I don't get what matrix they could be referring to. The process itself involves no matrices at all? It's just taking each vector in the set and subtracting the projection of the orthonormed basis being formed. repeat until basis is formed...
projection being from formula $$proj_w V = \frac{V \cdot a }{a^2} \vec{a} + \frac{V \cdot b }{b^2} \vec{b}$$
$$...$$ $$\left( \begin{array}{ccc} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} & 0 \\ \frac{4}{3 \sqrt{5}} & -\frac{2}{3 \sqrt{5}} & -\frac{\sqrt{5}}{3} \\ \end{array} \right)$$
Note that I got the correct orthonorm basis, I'm more concerned about what this B' B'' nonsense the question is referring to. Thanks for any clarification
I think the question is trying to ask the transformation matrix associated to different basis. As we know, every linear transformation can be associated a basis $M_{\epsilon\rightarrow\epsilon}(T)$. In this case, $T=P_W$ a projection linear transformation. Maybe you need to find $M_{B \rightarrow B}(T)$ and $M_{B'\rightarrow B'}(T)$, where $B'$ is your orthonormal basis.