Dention. A subset W of a vector space V is a subspace if
(1) W is non-empty
(2) For every v , w W and a, b F, av + bw W .
Expressions like av + bw, or more generally
ai v + i
are called linear combinations. So a non-empty subset of V is a
LECTURE 19: EXTERIOR PRODUCTS II
We ended last time looking at a basis for the exterior powers. Today we will
nish with that and describe the symmetric power. First we want a generalization
of the Reisz representation theorem.
Proposition. If f : U V W is
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LECTURE 22: INNER PRODUCTS
For the remaining lectures, we will work exclusively over R or C. Well also look
at a very special kind of bilinear form.
Denition 1. An inner product is a function , : V V F such that
(1) , 0 with equality i =
LECTURE 20: APPLICATIONS OF EXTERIOR PRODUCTS
Today will be a little dierent. Rather than focus too much on theory, we will
look at two very interesting examples involving the exterior product. But before
then, we have to talk a little more about functori
LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS
The last few lectures have focused on alternating multilinear functions. This one
will focus on symmetric multilinear functions. Recall that a multilinear function
f : U m V is symmetric if
LECTURE 23: METRIC SPACES
This lecture will look at the topological properties that arise from the notion of
distance provided by the norm.
Denition 1. If V is an inner product space, then the distance between and
d( , ) = | |.
PROBLEM SET #1
DUE SEPTEMBER 8
1. Find 2 distinct bases for Q2 .
2. Find all 3 bases of F2 .
3. Are the vectors (1, 1, 1), (0, 1, 2), (2, 1, 3), and (4, 0, 11) linearly independent?
If not, nd a dependence relation.
4. Let a, b F and let v, w V . Show t
LECTURE 28: ADJOINTS AND NORMAL OPERATORS
Todays lecture will tie linear operators into our study of Hilbert spaces and
discuss an important family of linear operators.
We start with a generalization of the Riesz representation theorem.
LECTURE 24: ORTHOGONALITY AND ISOMETRIES
Denition 1. If v and u are vectors in an inner product space V , then u and v
are orthogonal, written u v , if
u, v = 0.
Since V is an inner product space is orthogonal to is a symmetric relationship.
PROBLEM SET #2
DUE OCTOBER 6
1. Basic Properties of the Determinant. In this problem, let A be an n n
matrix over F.
(1) Using the denition, show that if A has two rows which are scalar multiples
of each other (so for some i and k , ai,j = ak,j for all j
LECTURE 17: PROPERTIES OF TENSOR PRODUCTS
Last time we showed how to build a universal vector space for bilinear forms.
Today we will look more at what this does for maps and what happens in more
We ended last time with the following theore
LECTURE 18: SYMMETRIC AND EXTERIOR PRODUCTS
We established in the last two lectures ways to identify bilinear maps from V W
to U with linear maps V W to U and ways to better understand the construction
of the tensor product. If we have extra properties of
Dention. A basis is a linearly independent spanning set.
Theorem. Every vector space has a basis.
We wont prove this; its actually essentially equivalent in the innite dimensional
case to one of the axioms of set theory: the axiom of ch
LECTURE 8 - DUALS CONTINUED
We saw last time that to any vector space V , we can naturally associate another
vector space, the dual space V of linear functional V F. This had the property
that if L : V W , then there is a linear map L : W V dened simply b
LECTURE 9 - EIGENVALUES
Having spent some time looking at the general properties of vector spaces and
the generic interplay with linear transformations, we turn our attention now to best
understanding a given linear operator on a given vector space. Thus
LECTURE 16: TENSOR PRODUCTS
We address an aesthetic concern raised by bilinear forms: the source of a bilinear
function is not a vector space. When doing linear algebra, we want to be able to
bring all of that machinery to bear. That means we need to cons
LECTURE : BILINEAR FORMS
Today we will focus on extra structure we can put on a vector space that render
certain constructions more natural.
Denition. A bilinear form , on V is a function
, : V V F
that is bilinear:
av + bu, w = a v , w + b u, w and
v , a