Page 110 - 35Linear Algebra

110 Vector Spaces

Propose definitions for addition and scalar multiplication in V . Identify
the zero vector in V , and check that every matrix in V has an additive
inverse.

5. Let P be the set of polynomials with real coefficients of degree three
R
3
or less.
(a) Propose a definition of addition and scalar multiplication to make
R
P a vector space.
3
(b) Identify the zero vector, and find the additive inverse for the vector
2
−3 − 2x + x .
(c) Show that P R is not a vector space over C. Propose a small
3
change to the definition of P 3 R to make it a vector space over C.
(Hint: Every little symbol in the the instructions for par (c) is
important!)

Hint

6. Let V = {x ∈ R|x > 0} =: R + . For x, y ∈ V and λ ∈ R, define
λ
x ⊕ y = xy , λ ⊗ x = x .
Show that (V, ⊕, ⊗, R) is a vector space.

7. The component in the ith row and jth column of a matrix can be
i
labeled m . In this sense a matrix is a function of a pair of integers.
j
S
For what set S is the set of 2 × 2 matrices the same as the set R ?
Generalize to other size matrices.
{∗,?,#}
8. Show that any function in R can be written as a sum of multiples
of the functions e ∗ , e ? , e # defined by
  
1 , k = ∗ 0 , k = ∗ 0 , k = ∗
e ∗ (k) = 0 , k = ? , e ? (k) = 1 , k = ? , e # (k) = 0 , k = ? .
0 , k = # 0 , k = # 1 , k = #
  
9. Let V be a vector space and S any set. Show that the set V S of all
functions S → V is a vector space. Hint: first decide upon a rule for
adding functions whose outputs are vectors.

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