Page 109 - 35Linear Algebra

5.3 Review Problems 109

5.3 Review Problems

Reading problems 1
Webwork:
Addition and inverse 2

x
2
1. Check that x, y ∈ R = R (with the usual addition and scalar
y
multiplication) satisfies all of the parts in the definition of a vector
space.

2
2. (a) Check that the complex numbers C = {x + iy | i = −1, x, y ∈ R},
satisfy all of the parts in the definition of a vector space over C.
Make sure you state carefully what your rules for vector addition
and scalar multiplication are.

(b) What would happen if you used R as the base field (try comparing
to problem 1).

3. (a) Consider the set of convergent sequences, with the same addi-
tion and scalar multiplication that we defined for the space of
sequences:

n o
N
V = f | f : N → R, lim f(n) ∈ R ⊂ R .
n→∞
Is this still a vector space? Explain why or why not.
(b) Now consider the set of divergent sequences, with the same addi-
tion and scalar multiplication as before:

n o
N
V = f | f : N → R, lim f(n) does not exist or is ± ∞ ⊂ R .
n→∞
Is this a vector space? Explain why or why not.

4. Consider the set of 2 × 4 matrices:

a b c d
V = a, b, c, d, e, f, g, h ∈ C
e f g h
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