MATH235 Study Guide - Midterm Guide: Brie, Diagonal Matrix, Orthogonal Matrix

• School: University of Waterloo
• Department: Mathematics
• Course: MATH235
• Professor: N A

Document Summary

Note: - only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks: short answer problems, de ne the four fundamental subspaces of a matrix a. Solution: let a be an m n matrix. Row(a) = {at (cid:126)x | (cid:126)x rm} Null(a) = {(cid:126)x rn | a(cid:126)x = (cid:126)0} Null(at ) = {(cid:126)x rm | at (cid:126)x = (cid:126)0: state the de nition of the rank of a linear mapping l : v w. Solution: rank(l) = dim (range(l)): let p and q be orthogonal matrices. Prove that p q is an orthogonal matrix. Solution: we have that p t p = i and qt q = i. Then (p q)t (p q) = qt p t p q = qt q = i. Hence, p q is an orthogonal matrix: give the de nition of an inner product (cid:104) ,(cid:105) on a vector space v.