Chapter 1. Intro of Vector Space

Question 1

Commutativity of addition in real vector space, R^n and C^n space.

Answer 1

u+v=v+u

Question 2

Associativity of addition in real vector space, R^n and C^n space.

Answer 2

(u+v)+w=u+(v+w)

Question 3

Associativity of multiplication in real vector space, R^n, and C^n space.

Answer 3

𝜆⋅（u⋅v)=(𝜆⋅u)⋅v

Question 4

Distributivity of multiplication in real vector space, R^n, and C^n space.

Answer 4

(𝜆+𝜇)⋅u=(𝜆⋅u)+(𝜇⋅u)

𝜆⋅(u+v)=𝜆⋅u+𝜆⋅v

Question 5

Multiplication identity in the calculation.

Answer 5

1⋅u=u

Question 6

Results of the calculation with 0.

Answer 6

u+0=u

0⋅u=0

Question 7

How do you define a Cartesian product? Suppose that A, B are sets.

Answer 7

We let A×B be the set of ordered pairs (a, b) with a∈A and b∈B.

Question 8

What is a real vector space given by?

Answer 8

1. A set V,
2. an element 0 of V,
3. a function V×V→V, (u, v)↦u+v, and
4. a function R×V→V, (𝜆, u)↦𝜆⋅u.

Question 9

What we call when (V, 0, +, ⋅) is a real vector space?

Answer 9

1. the elements of V vectors of the vector space (V, 0, +, ⋅),
2. 0 the zero vector of the vector space (V, 0, +, ⋅),
3. the function + the operation of addition of vectors,
4. the function ⋅ the operation of multiplication of vectors by real scalars.

Question 10

What is the first property of vector spaces?

Answer 10

Suppose that (V, 0, +, ⋅) is a real vector space. Then, for every u∈V, u+((-1)⋅u)=0.

Question 11

What is second property of vector spaces?

Answer 11

Suppose that (V, 0, +, ⋅) is a real vector space. Then, for every u, w∈V, if u+w=0, then w=-u.

Question 12

What is a complex vector space given by?

Answer 12

1. a set V,
2. an element 0 of V,
3. a function V×V→V, (u, v)↦u+v, and
4. a function C×V→V, (𝜆, u)↦𝜆⋅u.

Question 13

What are F-vector spaces?

Answer 13

We will use F to denote either R or C.
Consistently, we say that (V, 0, +, ⋅) is an F-vector space if it is either a real or complex vector space, depending on whether F=R or F=C.

Question 14

What is a zero matrix?

Answer 14

A zero n×d matrix over F is the n×d matrix over F whose entries are all equal to 0∈F.

Question 15

What are the vector spaces given by?

Answer 15

1. a set V,
2. an element 0 of V,
3. a function V×V→V, (u, v)↦u+v, and
4. a function C×V→V, (𝜆, u)↦𝜆⋅u.

Question 16

What are the sequences?

Answer 16

A sequence of elements of F is a function f:N→F.

Question 17

What is the third property of vector spaces?

Answer 17

Suppose that (V, 0, +, ⋅) is an F-vector space. Then, for every u∈V, -(-u)=u.

Question 18

What is the fourth property of vector spaces?

Answer 18

Suppose that (V, 0, +, ⋅) is an F-vector space. For every 𝜆⋅0=0.

Question 19

What is the fifth property of vector spaces?

Answer 19

Suppose that (V, 0, +, ⋅) is an F-vector space. Then, for every u∈V and 𝜆⋅0=0, then 𝜆=0 or u=0.

Question 20

What is the sixth property of vector spaces?

Answer 20

Suppose that (V, 0, +, ⋅) is an F-vector space, 𝜆∈F, and u∈V. The following assertions are equivalent:

1. 𝜆⋅u=0;
2. 𝜆=0 or u=0.

Question 21

What is the seventh property of vector spaces?

Answer 21

Suppose that (V, 0, +, ⋅) is an F-vector space. Then for every u∈V and n∈N,
u+u+...+u(n times)=n⋅u.

Question 22

What is the eighth property of vector spaces?

Answer 22

Suppose that (V, 0, +, ⋅) is an F-vector space, and u, v, w∈V are such that u+v=u+w. Then v=w.

Question 23

What is the ninth property of vector spaces?

Answer 23

Suppose that (V , 0, +, ·) is an F-vector space, and v ∈ V is such that u + v = u for every u ∈ V . (Hypothesis.) Then v = 0. (Thesis.)

Question 24

What properties a F-vector space should satisfies?

Answer 24

For every u, v, w ∈ V and λ, µ ∈ F:

1. u + v = v + u (commutativity of addition);
2. (u + v) + w = u + (v + w) (associativity of addition);
3. λ · (µ · u) = (λ · µ) · u ;
4. (λ + µ) · u = (λ · u) + (µ · u);
5. λ · (u + v) = (λ · u) + (λ · v);
6. u + 0 = u;
7. 0 · u = 0;
8. 1 · u = u.