Chapter 2. Vectors Space and Functions

Question 1

Give the definition of the addition of functions from F to F.

Answer 1

f+g∈F^F by setting (f+g)(x)=f(x)+g(x) for every x∈F.

Question 2

Give the definition of the multiplication of functions from F to F by scalars from F.

Answer 2

𝜆⋅f∈F^F by setting (𝜆⋅f)(x)=𝜆⋅f(x) for every x∈F.

Question 3

How to define a polynomial function?

Answer 3

A function f:F➝F is called a polynomial function over F if there exists n∈{0, 1, 2, …} and a0, a1, a2, …, an such that f is the function defined by f(x)=a0+a1⋅x+a2⋅x^2+…+an⋅x^n.

Question 4

What is the theorem of polynomial functions as a vector space?

Answer 4

We have that (P (F), 0, +, ·) is an F-vector space, where

1. P (F) is the set of polynomial functions over F
2. 0 is the zero function
3. • is the operation addition of polynomial functions over F
4. · is the operation of multiplication of polynomial functions over F by scalars from F.

Question 5

What is the definition of subspace?

Answer 5

Suppose that (V, 0, +, ·) is an F-vector space. An F-subspace of V (or, briefly, a subspace) is a nonempty subset U of V with the following properties:

1. for every x, y ∈ U, x + y ∈ U (here x + y is sum as vectors in V );
2. for every x ∈ U and λ ∈ F, λ · x ∈ U (here λ · x is scalar multiplication as a vector in V ).

Question 6

What is the theorem about Subspaces of the space of polynomial functions?

Answer 6

Suppose that a, b ∈ F. Set

U := {p ∈ P (F) : p (a) = b} . Then U is a subspace of P (F) if and only if b = 0.

Question 7

Suppose that A, B are set, f: A → B is a function, and A_0 is a subset of A. Then what is the definition of the restriction f | A_0?

Answer 7

The restriction f | A_0 is the assignment A_0 → B, a → f(a) obtained from f by restricting the domain from A to A_0 .