True-False for Quiz 6, Part B

These are just for practice and don't affect your grade. I'm assuming you've read over Leon, Ch 5.3 - 5.5. For this quiz assume that S and T are subspaces of Rⁿ, that A is an mxn matrix, and that b is a vector in Rⁿ.

True False. Rank (A^TA) = Rank A, and A^TA is nonsingular.

True False. If S Ç T = {0} then S Å T = Rⁿ.

True False. The columns of I form an orthonormal set.

True False. N(A^T) Å R(A) = R^m.

True False. If x^ is a least squares solution to Ax = b, then A(x^) = the projection of b onto R(A).

True False. If Ax = b is consistent, then any least squares solution is a solution ("a Ch.1 solution").

True False. The norm of a vector does not depend on the inner product used to define the norm.

True False. Every possible norm on V satisfies the triangle inequality.

True False. If Q is orthogonal, then Q^T = Q ^-1 , and the normal equations for Qx = b simplify to x^ = Q^Tb.

True False. If an inner product is used to define ||v||, then it will satisfy the Pythagorean law.

Written by S.Hudson