Assignment 6 – MATH 2210Q

Assignment 6 – MATH 2210Q
Due March 24th, 2016
Problem 1) In calculus II, we study real-valued sequences. For instance,
T = 1, 2, 4, 8, 16, . . .
and
1 1 1
U = 1, − , , − , . . .
3 5 7
are two real-valued sequences. A real-valued sequence is just an infinite sequence of real numbers. A usual
notation for a sequence S is S = (sk )k≥1 meaning that the k-th term of the sequence is sk . For instance,
for the sequence T above, we have tk = 2k−1 . Let V be the set of all real-valued sequences. Clearly, we
can add sequences in V (termwise addition) as follows:
(sk )k≥1 + (tk )k≥1 = (sk + tk )k≥1
and this gives a new sequence in V . For instance,
5 21
T + U = 2, , , . . .
3 5
Also, for c ∈ R and a sequence (sk )k≥1 , we can multiply each term of the sequence by c and define
(termwise scalar multiplication):
c(sk )k≥1 = (csk )k≥1
and this gives a new sequence in V . For instance,
3T = 3, 6, 12, 24, . . .
The
(a)
(b)
(c)
set V together with addition of sequences and scalar multiplication of sequences is a vector space.
What is the zero vector in V ?
What is the opposite (negative) of a sequence (sk )k≥1 ?
Prove that the subset U of those sequences that eventually become zero is a subspace of V .
Problem 2) Consider the following subsets U1 , U2 of R3 .
 2 




 x

 x+y

U2 =  y − x  | x, y ∈ R .
U1 =  x  | x, y ∈ R ,




y
2y
(a) Is U1 a subspace of R3 ? Explain.
(b) Is U2 a subspace of R3 ? Explain.
1
2
Problem 3) Consider the matrix


1
2 1
2
 2
0 1
1 

A=
 3
2 2
3 
1 −2 0 −1
(a) Find the null space of A and write your answer as the spanning set of linearly independent vectors.
(b) Find the column space of A and write your answer as the spanning set of vectors. Are these vectors
linearly independent? Why?
Problem 4) A linear transformation T : R2 → R2 is obtained as follows. Take ~x be an arbitrary vector
in R2 (that we think as a point in the plane). To get T (~x), we first project ~x onto the x-axis to get
P (~x), and then
x). For
√ we√apply a rotation of π/4 radians about the origin to P (~x) to get T (~
√
√ instance,
T ([1, 2]) = [ 2/2, 2/2] (here, P (~
x
)
=
[1,
0],
so
[1,
0]
is
rotated
about
the
origin
to
get
[
2/2,
2/2]). As
√
√
another example, T ([−2, 3]) = [− 2, − 2] (here, P (~x) = [−2, 0]).
(a) Describe the null space of T geometrically.
(b) Describe the range of T geometrically.
− − − − − − − − − − − − − − − − − − − − − − − − −−
Also, you should look at the following problems in the textbook. These problems are not to be handed in.
Section 4.1: Practice problems, and 1, 3, 5, 7, 9, 11, 21, 25, 29, 31.
Section 4.2: 1, 3, 5, 7, 11, 15, 17, 19, 23, 31, 33.