MAT223H1 Chapter 8.2: Chapter 8.2 Projection and the Gram-Schmidt Process
Document Summary
Let u and v be two vectors in r2 as depicted in the gure. Let u and v be vectors in rn, with v (cid:54)= 0, and let c be a nonzero scalar. Then (a) projvu is in the span{v}. (b) if u is in span{v}, then u = projvu (c) projvu = projcvu. Similar to the extension we did on the notion of orthogo- nality to subspaces, we can also de ne the projection of a vector onto a subspace. Let s be a nonzero subspace with orthogonal basis {v1, . The projection of u onto s, denoted by projsu is given by k(cid:88) i=1 projsu = vi u (cid:107)vi(cid:107)2 vi (2) Draw a line perpendicular to v that goes through the tip of u. The projection of u onto v, denoted projvu is the vector parallel to v at the intersection of v and the perpendicular line. k(cid:88)
