Chapter 2 Flashcards
Definition of span
We say that a list of vectors in a vector space V, spans V if every vector v element of V is a linear combination of the list of vectors
Definition of linear independence
A list of vectors in a vector space V is called linearly independent iff The equation ( kv1+ bla bla bal) = nul vector
Only has the trivial solution.
Definition of a basis
A list of vectors B= (e1, … , en) in a vector space V is called a basis for V iff it spans V and is linearly independent
Sifting lemma?
If a list of vectors spans a vector space V, then sifting the list will result a basis for V.
What is a coordinate vector?
Let B be a basis for a vector space V, and let v element of V,
[v]B = coln
Rules of Coordinate vectors?
Bl3
What is a change of basis matrix?
Bl3
Change of basis theorem?
Bl4
Linear Combination of preceding vectors, which statements are equivalent?
The list of vectors is linearly dependent.
Either v1=0 or for some r element of {2,3,…,n} vr is a linear combination of v1, v2,…, vr-1
Bumping off proposition
Bl6
Write down the basis for Polyn
Bl 7
Write down the basis for Trign
Bl7
Write down the basis for Matn,m
Bl7
Let W be a subspace of a finite dimensional vector space V, then W is finite-dimensional, and Dim(W)<= Dim(V)
Prove this
Bl8
Poly is infinite dimensional
Prove this
Bl9
