I have a question
Find the indicated projection matrix for the given subspace, and find the projection of the indicated vector $<2,-1,3>$ on $sp(<2,1,1>,<-\frac{8}{6},\frac{11}{6},\frac{5}{6}>)$ $ R^3$
I feel like there should be an easier way to get "missing" vectors when I need an orthonormal basis to get started on a question (since they seem to love putting these in the book despite it having nothing to do with the problem they're asking)
I do something like $$ \left( \begin{array}{ccc} 2 & \frac{8}{6} & x \\ 1 & \frac{11}{6} & y \\ 1 & \frac{5}{6} & z \end{array} \right) $$
and solve through dot products from the five equations I get. This is real time consuming and annoying, so I ask: is there a better way?
Given two vectors you can find a vector orthogonal to those two vectors by computing the cross product (see http://en.wikipedia.org/wiki/Cross_product), and more generally when you need to get an orthonormal basis start with some ordinary basis and apply the Gram-Schmidt process (see http://en.wikipedia.org/wiki/Gram-Schmidt_process)