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: