Vector Spaces, Subspaces and Bases

Question 1

Vector Space

Definition

Answer 1

-closed under scalar multiplication and vector addition

Question 2

Field

Definition

Answer 2

A system of numbers in which you can add, subtract, multiply and divide with the expected results
e.g. reals, complexes, rationals, F2

Question 3

Field

F2

Answer 3

F2 = {1,0}
like binary, 1+1=0
-1 = 1

Question 4

Axioms of a Vector Space

Answer 4

1) closure under addition and scalar multiplication
2) commutativity of addition
3) associativity of addition and scalar multiplication
4) identities for addition and scalar multiplication
5) inverses for addition
6) distributivity of scalar multiplication over addition

Question 5

Axioms of a Vector Space

closure under addition and scalar multiplication

Answer 5

v+w is defined for all v,w∈V

av is defined for all a∈F and for all v∈V

Question 6

Axioms of a Vector Space

commutativity of addition

Answer 6

v + w = w + v for all v,w∈V

Question 7

Axioms of a Vector Space

associativity of addition and scalar multiplication

Answer 7

u + (v + w) = (u + v) + w for all u,v,w ∈ V

(ab)v = a(bv) for all a,b∈F and for all v∈V

Question 8

Axioms of a Vector Space

identities for addition and scalar multiplication

Answer 8

there is an element 0∈V for which v + 0 = v for all v∈V

for the element 1∈F we have 1v=v for all v∈V

Question 9

Axioms of a Vector Space

Inverses for Addition

Answer 9

for every v∈V there is an element in V denoted (-v) for which v + (-v) = 0

Question 10

Axioms of a Vector Spacce

distributivity of scalar multiplication over addition

Answer 10

a(v+w) = av + aw for all a∈F and for all v,w∈V
(a+b)v = av + bv for all a,b∈F and for all v∈V

Question 11

Definition of Subtraction

Answer 11

we can define subtraction for vectors by defining u-v to be equal to u + (-v)

Question 12

Elementary Properties of Vector Spaces

Answer 12

i) if u=v+w then w=v-u
ii) Cancellation: if v+u=v+w then u=w
iii) if a∈F is any scalar and 0∈V is the zero vector then a0=0
iv) if 0 is the zero element of the field F and v∈V then 0v=0
v) if a∈F and v∈V and av=0 then either a=0 or v=0
vi) in any field F there are elements 0 and 1, and one can subtract scalars, so there is an element -1 in F. If v∈V then (-1)v = -v and in general (-a)v = -(av) for any a∈F

Question 13

Subspace

Definition

Answer 13

• let V be a subspace
• a subset, U, of V is called a subspace if:
  i) U is not empty
  ii) U is closed under vector addition
  iii) U is closed under scalar multiplication

Question 14

Properties of Subspaces

Answer 14

• any subspace of a vector space must contain the zero vector
• the intersection of two subspaces of V is also a subspace
• if U is a subspace of V then it becomes a vector space in its own right using the operations inherited from V
• if U is a subset of V which becomes a vector space in its own right using the operations inherited from V then U is a subspace of V

Question 15

Trivial Subspaces

Answer 15

• for any vector space V, the subset {|0} is a subspace of V called the trivial subspace
• the subset consisting of all V is also a subspace
• a proper subspace is one not equal to V

Question 16

Examples of Subspaces

Answer 16

• the subspaces of ℝ² are the trivial space, the whole space and lines through the origin
• the subspaces of ℝ³ are the trivial space, the whole space and lines and planes through the origin

Question 17

Span

Definition

Answer 17

-the 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 them so:
span(S) = {a1v1 + a2v2 +…+ anvn : a1,…,an∈F}

Question 18

What is the span of the empty set?

Answer 18

span(∅) = {0}

Question 19

Spans and Subspaces

Answer 19

• if S is a subset of the vector space V
• span S is a subspace of V
• it is the unique smallest subspace of V which contains v1, v2,…, vn, meaning:
• -S is a subset of spanS
• -given any other subspace U, if S is a subset of U then spanS is a subset of U

Question 20

Examples of Spans

Answer 20

i) if v is a non-zero vector in ℝ^n, then span{v}={av : a∈ℝ}

ii) a plane containing the origin in ℝ³ is the span of two vectors

Question 21

Spanning Set

Definition

Answer 21

-let V be a vector space and let S={v1, …,vn} be a finite subset of V.
-we say that S spans V or is the spanning set for V if every v∈V can be written as a linear combination of elements in S, i.e. if
v=a1v1+a2v2+…+anvn for some a1,a2,…,an∈F and for all v∈V

Question 22

Linear Independence

Definition

Answer 22

-let V be a vector space and let S={v1,v2,…,vn} be a finite subset of V
-we say that S is linearly independent if
a1v1+a2v2+…+anvn = 0
implies
a1=a2=…=an==0
-otherwise S is said to be linearly dependent

Question 23

Notes on Linear Independence

Answer 23

i) by convention the empty set is linearly independent
ii) any subset of a linearly independent set is linearly independent
iii) if some vi=0 then {v1,v2,…,vn} is linearly dependent
iv) {v} is linearly independent if and only if v≠0
v) {v,w} is linearly independent if and only if neither vector is a multiple of the other
vi) {v1,…,vn} is linearly dependent if and only if some vi is a linear combination of its predecessors

Question 24

Linear Independence

If and Only Ifs

Answer 24

• > {v1,..,vi,..,vj,..,vn} is LI <=> {v1,..,vj,..vi,..vn} is LI
• > {v1,..,vi,..vn} is LI <=> {v1,..avi,…,vn} is LI for a≠0
• > {v1,..,vi,..,vj,..,vn} is LI <=> {v1,..,vi,..,vj+avi,…,vn} is LI

Question 25

Operations on Matrices and Linear Independence

Answer 25

• Let B be obtained from A by row operations, then:
• > span of rows of A = span of rows of B
• > rows of A are linearly independent <=> rows of B are linearly independent

Question 26

How to find out if the rows of a matrix are linearly independent

Answer 26

-using row operations put the matrix into echelon form
-suppose B is an mxn over F in echelon form
-> rows of B span F <=> B has non-zero leading elements in every column
-> rows of B are linearly independent
B has no zero rows

Question 27

Basis

Definition

Answer 27

-let V be a vector space and S={v1,…,vn} a fininte subset of V
-S is a basis of V if:
->S spans V
-> the elements of S are linearly independent
-i.e. if every v∈V can be written in a UNIQUE way as a linear combination
v = a1v1 + a2v2 + …. +anvn

Question 28

Coordinate Vector

Definition

Answer 28

-let V be a vector space over F and let S={v1,v2,…,vn} be a basis of V
-if v∈V, the coordinate vector of v with respect to S is:
[v]s = |a = (a1, a2, a3,…,an)
with v=a1v1 + a2v2 +…+anvn

Question 29

Finite Dimensional

Definition

Answer 29

• if a basis of V has finite cardinality, V is finite dimensional
• if V is not finite dimensional, it is infinite dimensional

Question 30

What is the dimension of the empty set?

Answer 30

0

Question 31

Basis

IAOI Theorem

Answer 31

-the subset S={v1,…,vn} is a basis of V
<=>
-any element v∈V can be expressed UNIQUELY as a linear combination:
v=a1v1 + a2v2 +…+anvn

Question 32

Properties of Bases

Answer 32

• every vector space has a basis
• all bases of a vector space have the same number of elements
• dim(V) = cardinality of a basis of V
• linearly independent subsets {v1,…,vn} can be extended to a basis by adding certain other vectors of V
• linearly independent subsets in V have cardinality ≤ dim(V)
• any spanning set S={v1,…,vn} contains a basis of V, if S is not LI then certain vi can be removed to form a basis
• linearly independent subsets {v1,…,vn} of V can be extended to form bases by adding certain vectors of V
• LI sets in V have cardinality ≤ dimV

Question 33

Bases

Ifs

Answer 33

1) if U is a subspace of V and dim(U)=dim(V), then U=V
2) if S={v1,…,vn} is LI & dim(V)=n then S is a basis of V
3) if span{v1,…,vm}=V & dim(V)=n , then #{v1,…,vm}≥n i.e. cardinality of a spanning set is ≥ dimV
4) if span{v1,…,vn} & dimV=n, then [v1,…,vn} is a basis and LI

Question 34

Span

Equalities

Answer 34

span{v1,..,vi,..,vj,..,vn} = span{v1,…,vj,…,vi,…,vn}

span{v1,…,vi,…,vn) = span{v1,…,avi,….,vn} , a≠0

span{v1,…,vi,…,vj,…,vn} = span{v1,..,vi,..vj+avi,..,vn}

Question 35

Facts About Span

T and S

Answer 35

-let T={u1,…,um} and S={v1,…,vn} be subsets of V

• if T is a subset of spanS, then spanT is a subset of spanS
• spanT=spanS <=> T is a subset of spanS and S is a subset of spanT