Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 304 — Problem Set 11
Issued: 12.06
Due: training
11.1. Find the point on the line y = 2x that is closest to the point (5, 2).
11.2. Let ~x = (4, 4, −4, 4)> and ~y = (4, 2, 2, 1)> . Determine the angle between ~x and ~y .
11.3. Let S be the subspace of R3 spanned by the vector (1, −1, 1)> . Find
a basis of S ⊥ .
11.4. 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)
11.5. 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.
√ 

 


2/3
1
1/ √2





2/3
2 . Find the
11.6. Let ~u1 =
and ~u2 =
and ~x =
−1/ 2
1/3
2
0
projection p~ of ~x onto the span of ~u1 and ~u2 .
11.7. 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
11.8. Let A =  4
0
2
10
the column space of A. Find projection of ~b onto the column space.
11.9. Find distance from the point (1, 0, 0) to the set of solutions of the
system
x1 + 2x2 + x3 = 0
x1 + 3x2 − x3 = 0
11.10. Legendre Polynomials Consider the space C[−1, 1] of continuous
R1
functions on the interval [−1, 1] with the inner product hf |gi = −1 f (x)g(x) dx.
Use Gram-Schmidt orthogonalization to find an orthonormal basis of
the space of polynomials of degree at most two, by orthogonalizing the
basis {1, x, x2 }.