Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Summer 2015
Math 304 — Problem Set 5
Issued: 8.06
Due: training
5.1. Let ~x = (4, 4, −4, 4)> and ~y = (4, 2, 2, 1)> . Determine the angle between ~x and ~y .
5.2. Let S be the subspace of R3 spanned by the vector (1, −1, 1)> . Find
a basis of S ⊥ .
5.3. Which of the following sets of vectors form an orthonormal basis of
R2 ?
a) (1, 0)> , (0, 1)> ;
b) (3/5, 4/5)> , (5/13, 12/13)> ;
n √
√ > o
>
3/2, 1/2 , −1/2, 3/2
.
c)
5.4. Let {~u1 , ~u2 , ~u3 } be an orthonormal basis for an inner product space V
and let ~u = ~u1 + 2~u2 + 2~u3 and ~v = ~u1 + 7~u3 . Determine the values of
h~u|~v i, k~uk, k~v k.
√ 
 



1
2/3
1/ √2





2/3
2 . Find the
and ~x =
5.5. Let ~u1 =
and ~u2 =
−1/ 2
2
1/3
0
projection p~ of ~x onto the span of ~u1 and ~u2 .
5.6. Consider the space C[0, 2π] of continuous
R 2π functions on the interval
[0, 2π] with the inner product hf, gi = 0 f (t)g(t) dt. Find projection
of the function f (x) = 2x onto the subspace spanned by {1, cos x, sin x}.




3 −1
0
2 , and ~b =  20 . Find an orthonormal basis of
5.7. Let A =  4
0
2
10
the column space of A. Find projection of ~b onto the column space.
5.8. Find distance from the point (1, 0, 0) to the set of solutions of the
system
x1 + 2x2 + x3 = 0
x1 + 3x2 − x3 = 0
5.9. Consider the space C[−1, 1] of continuous
R 1 2functions on the interval
[−1, 1] with the inner product hf |gi = −1 x f (x)g(x) dx. Use GramSchmidt orthogonalization to find an orthonormal basis of the space
of polynomials of degree at most two, by orthogonalizing the basis
{1, x, x2 }.


1 2 −1
2 .
5.10. Find characteristic polynomial of the matrix A =  0 1
0 2
1
1 1
5.11. Is the matrix A =
diagonalizable?
−1 3
5.12. In each of the following, factor the matrix A into a product T DT −1 ,
where D is diagonal.
5
6
(a) A =
;
−2 −2
2 −8
(b) A =
;
1 −4


1 0
0
3 .
(c) A =  −2 1
1 1 −1


1
1
1
5.13. Compute etA for A =  −1 −1 −1 .
1
1
1
√
and the formula for a function of a
5.14. Find A usingdiagonalization
2
1
matrix, for A =
.
−2 −1