advertisement

Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000 Quiz 8 – due Monday, October 11, 1999 Box # NAME You may use the handouts, the notes, your notes, Maple, and you may work with other students, but each student must hand in a quiz. Grades will be 0/10 or 10/10 – absolutely nothing in between, and if anything is wrong the grade is 0. Let A be a 3 £ 2 matrix with rank 2, and suppose that Ax = b has no solution. (a) dim(N (A)) = (b) dim(R(A)) = (c) Geometrically, R(A) is a (circle the correct answer) point, line or plane in R3 : (d) T or F b 2 R(A): (e) T or F b 2 R3 : (f) T or F The normal equations are A> Ax = A> b: (g) T or F x 2 R2 : If x is the least squares solution (i.e. the solution to A> Ax = A> b); then (h) T or F If x is the least squares solution, then Ax 2 R3 : (i) T or F If x is the least squares solution, then Ax 2 R(A): Give the geometric sketch of A: Be sure to include (and label), N (A); R(A); b; x (the least squares solution), Ax; and p = the projection of b onto the range of A: