Chapter 1

Question 1

Examples of fields?

Answer 1

• real numbers
• complex numbers
• integers modulo n

Question 2

What is a field?

Answer 2

A set with the operators addition and multiplication

Question 3

What 5 axioms do operators of a field have to satisfy?

Answer 3

F1) Commutativity

F2) Associativity

F3) Distributivity

F4) Identities

F5) Inverses

Question 4

What is the field of 2 numbers called?

Answer 4

Modulo 2

Question 5

What would a vector in (F₂)³ look like?

Answer 5

(1̅, 0̅, 1̅)

Question 6

What is a vector space?

Answer 6

A vector space V over a field F is a set with 2 operations: addition and multiplication by scalars

Question 7

What 5 axioms do operators of a vector space have to satisfy?

Answer 7

V1) Closure

V2) Commutativity and associativity of addition

V3) Distributivity and compatibility of scalar multiplication

V4) 1v = v

V5) There is a unique element, call it 𝟎 ⋲ V, such that 0𝐯 = 𝟎

Question 8

What is a subspace?

Answer 8

A subspace of V is a subset U which contains 𝟎 and is closed under addition and scalar multiplication

Question 9

What do you need to do to show something is not a subspace?

Answer 9

Find a counter-example

Question 10

What is a linear span?

Answer 10

A linear span of a finite set of vectors S = {v₁, v₂, …, vn} in a vector space V is the set of all linear combinations of the vectors. That is:

span(S) = {ɑ₁v₁ + … + ɑnvn : ɑ₁ … ⋲ F}

Question 11

How would you prove span(S) is a subspace of V? where S = {v₁, v₂, …, vn} where vi ⋲ V∀i

Answer 11

Check that:

1. Contains zero element
2. Addition holds
3. Scalar multiplication holds

Question 12

What are the 3 different types of subspace of R²?

Answer 12

• zero vector
• lines through the origin ( = R¹)
• planes through the origin ( = R²)

Question 13

What is a spanning set?

Answer 13

S = {v₁, v₂, …, vn} is a spanning set of V if span(S) = V

we say that S spans V

Question 14

What is span(2u) equal to?

Answer 14

span(u)

Question 15

What is span(u₁, u₁ + u₂) equal to?

Answer 15

span(u₁, u₂)

Question 16

What is span(u₁, u₂, u₂) equal to?

Answer 16

span(u₁, u₂)

Question 17

What is linear dependence?

Answer 17

A set of vectors {𝐯₁, 𝐯₂, …, 𝐯n} are linearly dependent if

ɑ₁𝐯₁ + ɑ₂𝐯₂ + … + ɑn𝐯n = 𝟎 for some non zero a_is

Question 18

What is a basis?

Answer 18

Let V be a vector space.

A finite set of vectors S = {𝐮₁, 𝐮₂, …, 𝐮n} is a basis of V if it spans V and is linearly independent

Question 19

What is equivalent to the statement ‘S = {𝐮₁, 𝐮₂, …, 𝐮n} is a basis of V’?

Answer 19

Every 𝐮 ⋲ V can be uniquely written as 𝐮 = ɑ₁𝐮₁ + ɑ₂𝐮₂ + … + ɑn𝐮n

Question 20

What is a bijection?

Answer 20

A 1-1 map

Question 21

What is the definition of dimension?

Answer 21

If V has a basis with n elements, then V has dimension n, dim(V) = n

Question 22

What is the Steinitz exchange lemma?

Answer 22

Let {𝐯₁, …, 𝐯n} span V, and {𝐮₁, …, 𝐮m} be a linearly independent subset of V. Then m < n and the set {𝐮₁, …, 𝐮m, 𝐯(m₊₁), …, 𝐯n} span V

(though you might have to reorder the 𝐯is)

Question 23

What method to prove Steinitz exchange lemma?

Answer 23

Proof by induction

Question 24

What is the relationship between the no. of elements in a LI set and the no. of elements in a spanning set?

Answer 24

No. of elements in LI set < spanning set

Question 25

Do any 2 bases of V have the same number of elements?

Answer 25

Yes

Question 26

If dimV = n, and you have n vectors in V, when will they form a basis for V?

Answer 26

If these span V or if they are linearly independent

Question 27

When are the definitions for span, linear independence and bases equivalent?

Answer 27

For finite-dimensional vector spaces

Question 28

W is a subset of V and dim(V) = n. What does this mean for dim(W)?

Answer 28

• dim(W) <= dim(V)

* if dim(W) = dim(V) then W=V

Question 29

If matrix B can be obtained from matrix A by row operations, what implications does this have? (2 points)

Answer 29

1. span of rows of A = spans of rows of B

2. rows of A are LD ⤄ rows of B are LD

Question 30

Does every spanning set of V contain a basis for V?

Answer 30

Yes

Question 31

Examples of commutativity? using a, b ⋲ F

Answer 31

• a + b = b + a

* a • b = b • a

Question 32

Examples of associativity? using a, b, c ⋲ F

Answer 32

• a + (b + c) = (a + b) + c

* a • (b • c) = (a • b) • c

Question 33

Example of distributivity? using a, b, c ⋲ F

Answer 33

a • (b + c) = a • b + a • c