lecture 7 - vector spaces Flashcards
(13 cards)
What are the axioms for a vector space?
u + v = v + u
u + (v + w) = (u + v) + w
u + O = u (neutral element)
u + (-u) = O
scalar = c
c(u + v) = cu + cv
scalar c, p
(c + p)u = cu + pu
+ 2 closures
what is a vector space?
a collection of vectors that follow the vector axioms.
- can be represented in lines, numbers (e.g. (1,2,3)
analogy: huge invisible grid where vectors allow movement
anywhere inside the space.
what are the 2 closures
vector addition = if u and v are in space E, then u + v must also be in the space E.
homogeniality (scalar multiplication) = if v is in space and scalar C is an element of set of REAL (or neutral) numbers, then cv is in space.
what is a subspace?
subspace (U,+, .) is a subspace of larger vector (E, +, .), following the same rules of addition and scalar multiplication.
Must meet:
- includes zero vector
- closed under addition
- closed under homogeniality
e.g. The xy-plane {(𝑥,𝑦,0)}, is a subspace of 𝑅^3 because any sum or scalar multiple of its vectors stays in the plane.
what vector is always part of a subspace?
the zero vector
how to check if a set is a vector space?
- check closures
> addition if u,v in set, u + v is too
> homogeniality if v in set cv is too - check zero vector:
- does set include 0 (required for vector space) - check vector axioms
> commutativity u + v = v + u
> associativity (u + v) + w = u + (v + w)
> distributive and identitiy
! If set fails closure or is missing the zero vector, then its not a vector space!
What is the intersection of two vector spaces?
set of all vectors that belong to both vector spaces U and V.
Method:
1. solve the system of equations defining U and V together.
2. Express solution in terms of the span of vectors.
Example:If U={(x,y,z)∣x+y−2z=0} and
V={(x,y,z)∣4x−y−2z=0}, solve both equations to find common vectors.
! = U intersection V is always a subspace of U and V.
What is the sum of two vector spaces?
U + V
set of all vectors that can be written as u + v, where u is element of U and v is element of V vector spaces.
Method:
1. find basis vectors U and V
2. combine all basis vectors
3. Remove dependent vectors (if necessary)
Example: If U=span{(1,0,0)} and V=span{(0,1,0)}, then
U+V=span{(1,0,0),(0,1,0)}
which is a plane in R^3
! = U + V is the smallest vector space containing U and V
What is a linear span?
Set of all possible linear combinations of given vectors.
Written L(v1, v2,…, vn)
L{v1, v2,…, vn}={a1v1 + a2v2 +⋯+ anvn
∣ai∈R}
Eg. (1,0,1) and (1,0,0) span a plane inside of R^3 to make span of L( (1,0,1), (1,0,0)) = plane.
! Span of a set of vectors is always a subspace!
What do E + and . mean in the triple?
E = non-null set
+ internal composition of E X E > E
. external composition of R X E > E
how to solve intersection?
- write down equations of space
- solve simultaneously (like system of equations)
- express result as a span of a set of vectors
how to solve sum of vectors?
- find spanning set for each space( in terms of basis vectors)
- combine all basis vectors from both spaces
- check for linear dependence (remove redundancy if needed)
> if linearly independent, span is plane
> if linearly dependent, U + V is line
what is the relation between linear span and rank?
the no. of linearly independent vectors is equal to the rank of those vectors in a matrix.
in linear span, you can remove some linearly dependent vectors to keep only the independent vectors which can then relate to the rank (rho) of the matrix.
