Space
Vector Axioms
The kernel of T is {c Rn T (c) = 0}
(a) Vector Addition
(e) Additive Identity
(b) Scalar Multiplication
(f) Multiplicative Assoc.
(c) Additive
tivity
(g) Multiplicative
tity
Commuta-
(d) Additive Associativity
Iden-
b
a
Bases and Dimension
1
=
(b) L2 = f
2
= a f (x) 2 dx
b
B is the basis for V if B spans V and B is linearly
independent.
f (x) dx
(a) L = f
(c) L = f
You can have k linearly independent vectors in nspace, such that 0 k n. (Work in class and on
hws from 3.76)
(h) Proper Distribution
L-Norms for C [a, b]
1
K + 1 vectors in K -space are linearly dependent.
(Theorem 3.76)
If B is independent then every vector x V can
uniquely be represented by the elements of B .
12
= maxx[a,b] f (x)
Inner Product for C [a, b] - < f, g >=
b
a f (x)g(x)dx
Subspace
With vector space V and a subset, S , of V , S is a
subspace of V if S is also a vector space. To show
this S must be closed under vector addition, scalar
multiplication and be nonempty. (Theorem 3.14)
If S and T are subspaces of a vector space of V then
S T is a subspace of V . (Lemma 3.25)
Spans
Denition - The span of a set A is the set S of all
the possible linear combinations of the elements of
A. span{x1 , , xn } = {n=1 ck xk R}
k
Linear Denition Independency
- A set of vectors is linearly independent if
there is no nontrivial linear combination that equals
zero.
If S = {x1, , xn } is a set of nitely many vectors
in V . And dene a function T Rn V, T (c) =
n=1 ck xk Then T is injective IFF S is linearly indek
pendent.
The above T is surjective IFF S spans V .
For nite B that is a basis for V , all other bases for
V must have the same number of elements as B .
V is a nite vector space. If {x1 , , xk } are independent, but dont span V , then there are vectors
{xk+1 , , xn }so that B = {x1 , , xk , xk+1 , , xn } is a
basis for V .
If 2 are true all 3 are. B is independent, B spans V ,
and the number of vectors in B = dim(V ).
B is a set of nitely many independent vectors in V .
If no larger set contains B and is independent, then
B is a basis for V .
A minimal spanning set is a basis (ex. 3.94)
Every spanning set contains a basis. (ex. 3.95)
Denition - [x]B = (c1 , , cn ) that is, the components of x then make a component vector for x.
Random things from class
Pn is the set of all polynomials with degree at most
n.
F (R) is the set of all functions with real number
coecients.
The empty set is independent, but {0} is dependent.
e1 = (1, 0, ), e2 = (0, 1, 0, ) form the standard basis for Rn