Vector Spaces/subspaces Flashcards
What is a vector space and what properties does it have
A vector space is an additive group with property of linearity
What are conditions for a subset U of V to be a subspace of V
Conditions for U to be a linear subspace of V are:
0 element of U
U is closed under linear combinations, v,w element of U, av + bw element of U
What is a linear map psi
A linear map phi is if psi(av+bw)= apsi(v) + bpsi (W)
What are examples of linear maps
Examples of linear maps are:
Id: V to V
If psi is linear, psi(0)=0 and psi(-v)=-psi(v)
If phi and psi are linear, then psi o phi is linear
If psi is a linear bijection, inverse psi is linear
What is an (linear) isomorphism
An isomorphism is a linear bijection V to W, we say V is isomorphic to W
What is the kernel of a linear map V to W
Kernel of a linear map is:
Ker(phi)= for all v element of V (phi(v)=0)
What is the Im(phi) V to W
Im(phi) is w = phi(v) = column space (span of n columns) of matrix A representing phi
If phi V to W is a linear map, what are ker(phi) and Im(phi)
Ker(phi) is a linear subspace of V
Im(phi) is a linear subspace of W
What is linear span for a subset S of V
Linear span for a subset S of V is:
All linear combinations of v1,…,vN element of S
Sigma i=1 to n, Livi where Li element of the field
When is a set of vectors linearly independent
A set of vectors is linearly independent if they can’t be written as linear combinations of each other.
Sigma i=1 to n Livi =0 implies Li=0 for all i
When is a list of vectors in V spanning
A list of vectors in V is spanning if the list of vectors span V, so all vectors in V can be acquired using linear combinations of the list of vectors.
When is a list of vectors a basis
A list of vectors is a basis if it’s both linearly independent and spanning.
What is an example of a basis not being unique
An Example of a basis not unique is:
E1,e2 is a basis of R^2, e1+e2, e2 is also a basis
