UCSC MATH 21 FALL 2007
Review Questions 5
1. Find (a basis for) the orthogonal complement of the subspace V⊂R4that is spanned by the
vectors
1
2
0
3
and
2
1
1
4
.
2. Find an orthonormal basis for the subspace Vin problem 1.
3. Find the matrix of the projection onto the subspace Vin problem 1.
4. What is the rank of the matrix from problem 3? Explain how to answer this question without
actually looking at the matrix itself.
5. True (with a brief explanation) or False (with a simple counterexample):
a. If Uis an n×northogonal matrix, and {x1,...,xn}is an orthonormal basis of Rn, then
{Ux1, . . . , U xn}is also an orthonormal basis of Rn.
b. If Uis an n×northogonal matrix, and {v1,...,vm}is an orthonormal basis of the subspace
V⊂Rn, then {Uv1, . . . , U vm}is also an orthonormal basis of V.
c. If Aand Bare symmetric n×nmatrices, then AB is symmetric.
d. If Ais an n×mmatrix, then AATis a symmetric matrix.
e. If Bis an n×mmatrix whose columns form an orthonormal set, then BBTis an orthogonal
matrix.
f. If Ais an n×mmatrix and ker(A) = {0}, then ATAis invertible.
6. Find the least-squares (approximate) solution of the equation
210
131
024
311
·
x
y
z
=
1
1
1
1
.
7. Find the coefficients aand bof the line y=ax +bthat minimize the sum of squares
4
X
j=1
(axj+b)−yj2
,
for the points (x1, y1) = (1,1), (x2, y2) = (2,3), (x3, y3) = (3,4) and (x4, y4) = (4,2).
1