4) Vector Spaces and Bases

Question 1

What is a vector space

Answer 1

Question 2

Describe the proof that For any vector for any vector v ∈ V , there is a unique u ∈ V such that v + u = 0

Answer 2

This proves the uniqueness of u.

Question 3

What is a subspace of a vector space

Answer 3

A subset U ⊆ V is a subspace of V if:
* U≠ ∅.
* For all u, v ∈ V ,u, v ∈ U ⇒ u + v ∈ U.
* For all u ∈ V and λ ∈ K, u ∈ U ⇒ λu ∈ U

Question 4

What is a linear combination

Answer 4

Question 5

Under what condition is a subset of a vector space a subspace

Answer 5

If it is closed under linear combinations
This means that,
* * u ∈ U and λ ∈ K, λu ∈ U, so U is closed under scalar multiplication
* For any u, v ∈ U, u + v = 1u + 1v ∈ U so U is closed under addition

Question 6

What is the span of a subset

Answer 6

The span of a non-empty subset S ⊆ V is the set of all linear combinations of vectors from S

Question 7

What is the span of the empty set

Answer 7

Span(∅) = {0}

Question 8

What is Linear Independence

Answer 8

Question 9

What is a Spanning Set

Answer 9

S is a spanning set of V if V = Span(S)

Question 10

What is a basis

Answer 10

A set of vectors, B ⊆ V , is a basis of V if it is a linearly independent spanning set of V

Question 11

What is the standard basis of the complex numbers

Answer 11

{1,i}

Question 12

What is the standard basis of Pn (the polynomial set)

Answer 12

Question 13

What does a finite dimensional vector space mean

Answer 13

It has a finite basis

Question 14

What is the relationship between a finite spanning set and a linearly independent subspace

Answer 14

• Let V be a vector space and let S ⊆ V be a finite spanning set.
• Let L ⊆ V be linearly independent
• Then L is finite and |L| ⩽ |S|

Question 15

What is the Basis Theorem

Answer 15

Let V be a finite-dimensional vector space. If B and C both are bases of V then B and C are finite sets and |B| = |C|

Question 16

What is the Proof of the Basis Theorem

Answer 16

• Let B be a linearly independent set and C be a spanning set
• This implies that |B| ⩽ |C|.
• Since B is a spanning set and C is a linearly independent set this implies that |C| ⩽ |B|. Therefore, |C| = |B|

Question 17

What is the dimension of a vector space

Answer 17

Let V be a finite-dimensional vector space, with basis B. The dimension of V is dim V = |B|

Question 18

What is the dimension of the vector space Mmn(K)

Answer 18

dim Mmn(K) = mn

Question 19

What is the dimension of Pn (the polynomial set)
{1, x, x2, . . . , xn}

Answer 19

dim Pn = n + 1

Question 20

What is the relationship between a subset of the vector space, a basis of the vector space and the spanning set of a vector space

Answer 20

• Let V be a finite-dimensional vector space
• Let L ⊆ V be a linearly independent subset
• Lt S ⊆ V be a spanning set of V such that L ⊆ S.
• There must be a basis B such that L ⊆ B ⊆ S.

Question 21

If V is a vector space and u ∈ V such that u ∉ L, what does this tell us

Answer 21

u ∉ Span(L) ⇐⇒ L ∪ {u} is linearly independent

Question 22

Describe the proof that u ∉ Span(L) ⇐⇒ L ∪ {u} is linearly independent (Don’t need to know)

Answer 22

Question 23

What is the condition for subspace or spanning set implies it is actually a basis

Answer 23

• Let V be a finite-dimensional vector space and let L ⊆ V be a linearly independent subset
• Then |L| ⩽ dim V and, if |L| = dim V then L is a basis of V .
  Or
• Let V be a vector space and let S ⊆ V be a finite spanning set.
• Then V is finite-dimensional, |S| ⩾ dim V , and if |S| = dim V then S is a basis.

Question 24

If U ⊆ V and dim U = dim V what does this imply about U and V

Answer 24

U = V

Question 25

Describe the proof that if U ⊆ V, if dim U = dim V then U = V (Don’t need to know)

Answer 25

• Any linearly independent subset of U is also linearly independent as a subset of V , so all linearly independent subsets of U are finite and have cardinality at most dim V
• Let B be a linearly independent subset of U which is of largest possible cardinality
• Assume for contradiction that B is not a spanning set of U. There is a vector
  u ∈ U such that u ̸∈ Span(B), and therefore B * ′ = B ∪ {u} is linearly dependent
• But |B′ | > |B|, and this contradicts our choice of B. Therefore, B is a
• spanning set of U, so it is a basis of U.
• Since B is a linearly independent subset of V , there is a basis C of V such that B ⊆ C. We have dim U = |B| ⩽ |C| = dim V.
• If dim U = dim V then B = C, and so U = Span(B) = Span(C) = V .

Question 26

What is the sum of vector spaces

Answer 26

For any two subspaces U and W of a vector space V , we define their
sum U + W = {u + w | u ∈ U, w ∈ W}

Question 27

What is the intersection of vector spaces

Answer 27

U1 ∩ U2 ∩ · · · ∩ Un

Question 28

For subspaces U and W of a finite-dimensional vector space V ,
what is the dim(U + W)

Answer 28

dim(U + W) = dim U + dim W − dim(U ∩ W).

Question 29

Describe the proof that dim(U + W) = dim U + dim W − dim(U ∩ W) (Don’t need to know)

Question 30

What is the dim(U + W) if U ∩ W = {0}

Answer 30

dim(U + W) = dim U + dim W

Question 31

What is the coordinate vector of v with respect to b

Answer 31

Question 32

When are u and v equal to each other (in terms of coordinate vectors)

Answer 32

u = v ⇐⇒ [u]B = [v]B.
Where B is the basis of V

Question 33

What is the change of basis matrix from B to C

Answer 33

Question 34

Is the change of basis matrix unique

Answer 34

Question 35

Describe the proof of when the change of basis is unique (Don’t need to know)

Answer 35

Question 36

What are

Answer 36

Question 37

Describe the proof of

Answer 37

Question 38

What is equivalent to a matrix being invertiable (Updated Version)

Answer 38