Vector Space
Definition
-closed under scalar multiplication and vector addition
Field
Definition
A system of numbers in which you can add, subtract, multiply and divide with the expected results
e.g. reals, complexes, rationals, F2
Field
F2
F2 = {1,0}
like binary, 1+1=0
-1 = 1
Axioms of a Vector Space
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
Axioms of a Vector Space
closure under addition and scalar multiplication
v+w is defined for all v,w∈V
av is defined for all a∈F and for all v∈V
Axioms of a Vector Space
commutativity of addition
v + w = w + v for all v,w∈V
Axioms of a Vector Space
associativity of addition and scalar multiplication
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
Axioms of a Vector Space
identities for addition and scalar multiplication
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
Axioms of a Vector Space
Inverses for Addition
for every v∈V there is an element in V denoted (-v) for which v + (-v) = 0
Axioms of a Vector Spacce
distributivity of scalar multiplication over addition
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
Definition of Subtraction
we can define subtraction for vectors by defining u-v to be equal to u + (-v)
Elementary Properties of Vector Spaces
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
Subspace
Definition
- 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
Properties of Subspaces
- 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
Trivial Subspaces
- 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
Examples of Subspaces
- 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
Span
Definition
-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}
What is the span of the empty set?
span(∅) = {0}
Spans and Subspaces
- 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
Examples of Spans
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
Spanning Set
Definition
-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
Linear Independence
Definition
-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
Notes on Linear Independence
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
Linear Independence
If and Only Ifs
- > {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
