Question

LINEAR ALGEBRA: Find the projection of the vector v = (7,0,1) onto the subspace W spanned by (1,2,-1) and (-1,1,1)

Answer 1

@amistre64 @surjithayer

Answer 2

First, note that your two basis vectors are orthogonal (dot product zero). Then\[v^*=\frac{<v,u>}{<v,v>}u+\frac{<v,w>}{<w,w>}w\]Where v* is the projection of v into the space spanned by u and w.

Answer 3

Inner product in this case is the familiar dot product. That gives us a scalar times a vector plus a scalar times a vector.

Answer 4

If we take u to be the first spanning vector, and w the second, I think it works out to something like \[v^*=u-2w\]

Answer 5

If you are interested, a little substitution and vector math will put it into terms of vectors i, j, and k.