A First Course in Linear Algebra, 2017a

9.1. Algebraic Considerations 463
♠
An important use of the additive inverse is the following theorem.
Theorem 9.10:
Let V be a vector space. Then⃗v + ⃗w =⃗v +⃗z implies that ⃗w =⃗z for all⃗v,⃗w,⃗z ∈ V
The proof follows from the vector space axioms, in particular the existence of an additive inverse (−⃗v).
The proof is left as an exercise to the reader.
Exercises
Exercise 9.1.1 Suppose you have R 2 and the + operation is as follows:
(a,b)+(c,d)=(a + d,b + c).
Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.
Exercise 9.1.2 Suppose you have R 2 and the + operation is defined as follows.
(a,b)+(c,d)=(0,b + d)
Scalar multiplication is defined in the usual way. Is this a vector space? Explain why or why not.
Exercise 9.1.3 Suppose you have R 2 and scalar multiplication is defined as c(a,b)=(a,cb) while vector
addition is defined as usual. Is this a vector space? Explain why or why not.
Exercise 9.1.4 Suppose you have R 2 and the + operation is defined as follows.
(a,b)+(c,d)=(a − c,b − d)
Scalar multiplication is same as usual. Is this a vector space? Explain why or why not.
Exercise 9.1.5 Consider all the functions defined on a non empty set which have values in R. Isthisa
vector space? Explain. The operations are defined as follows. Here f ,g signify functions and a is a scalar.
( f + g)(x) = f (x)+g(x)
(af)(x) = a( f (x))
Exercise 9.1.6 Denote by R N the set of real valued sequences. For⃗a ≡{a n } ∞ n=1 ,⃗ b ≡{b n } ∞ n=1 two of these,
define their sum to be given by
⃗a +⃗b = {a n + b n } ∞ n=1
and define scalar multiplication by
c⃗a = {ca n } ∞ n=1 where ⃗a = {a n} ∞ n=1