M1-2A

def: linear combination

a list of vectors v1,…,vn in V of the form a1v1+…+anvn where a in F

def: span

the set of all linear combinations of a list of vectors v1,…,vn in V is denoted as span(v1,…,vn)

thm: span is the smallest containing subspace

the span of a list of vectors in V is the smallest subset of V containing all vectors in the list

def: spans

If span(v1,…,vn)=V then we say the list v1,…,vn spans V

def: finite-dim vector space

V is finite-dim if some list of vectors spans the space

def: polynomial P(F)

a function p from F to F is a polynomial with coefficients in F such that p(z)=a0+a1z+…+amz^m

def: degree of a polynomial

a polynomial p has degree m if p(z)=a0+a1z+…+amz^m and am=!0

def: Pm(F)

the set of all polynomials with coefficients in F and degree at most m

def: infinite-dim vector space

V is infinite-dim if its not finite-dim

def: linearly independent

v1,…,vn in V is linearly independent if the only way from a1v1+…+amvm=0 is for all a=0

def: linearly dependent

v1,…,vn in V is dependent if a1va+…+anvn=0 is true for not all a=0

thm: linear dependence lemma

suppose v1,…,vn is dependent then there exists a vk in the span (v1,…,vk-1) and if the kth term is removed from v1,…,vn the span of the remaining list equals the span(v1,…,vn)

thm: length of independent list <= length of spanning list

in a finite-dim vector space the length of every independent list of vectors is less than or equal to the length of every spanning list

thm: finite-dim subspaces

every subspace of a finite-dim vector space is finite-dim