Albegra 1B Flashcards
When is V finite dimensional
V is finite dimensional if V has a finite list of vectors as a basis
What is dim V
Dim V is number of L.I vectors in basis
When is a list of vectors a basis
A list of vectors is a basis when any v in V can be written as a linear combination of this list
What is the standard basis
Standard basis is:
Ei in F^I,
Ei(j) = 1 if I =j, 0 otherwise
What are examples of standard basis
Examples of standard bases are:
F^(nx1) Ei = (0,,,1,,,0) 1 in the ith row
Dim M x N matrices is M*N
What is a linear map psi
A linear map psi is when psi preserves addition and scalar multiplication
Psi(v+w) = psi(v) + psi(w)
Psi(Lv) = Lpsi(v)
What is kernel of psi V to W
Kernel of psi is V to W is:
All v in V s.t psi(v) = 0 (subspace of V)
What is image of a map psi V to W
Image of a map psi V to W is psi(v) (all v in V), subspace of W
When is a map psi linear
A map psi is linear when:
Psi(v+Lw) = psi(v) + Lpsi(w)
What are examples of linear maps
Examples of linear maps are:
ID(v) : V to V is linear
If phi and psi are linear, psi o phi is linear
What is an isomorphism
An isomorphism is a kinear map phi V to W if exists map psi : W to V s.t psi o phi = id(v) and vice versa. Phi is isomorphic iff bijective
What is a way to tell if something isn’t a linear map
Way to tell if something isn’t linear map is:
Phi(v) =/ 0 phi(v + -v)
