call # 134202
MATH 601, MWF 9:30 a.m., SO 0056
DETAILED LIST OF TOPICS
Sept. 22:
spaces IK
n
, function spaces, subspaces, ﬁnite sets of vectors as opposed to unordered
systems and ordered systems of vectors, span of a system, spanning systems
Sept. 24:
DIMENSION, inﬁnite dimensionality of the space of polynomials, linear
(in)dependence, elementary modiﬁcations, the replacement theorem
Sept. 26:
BASIS, the algorithm for selecting a basis from a spanning system, extending
a linearly independent system in a ﬁnitedimensional space to a basis, the dimension of a
subspace, the case where the space and a subspace are of the same dimension
Sept. 29:
LINEAR OPERATORS, endomorphisms, scalar operators, matrix operators,
the derivative as a linear operator, the operator with prescribed values for a ﬁxed basis,
linear functionals (dual vectors), Dirac deltas (evaluation functionals), deﬁnite integration
as a functional, the spaces
L
(
V,W
) and End
V
, the dual space
V
*
, composition
Oct. 1:
COMPONENTS of a vector relative to a basis, the index notation and summing
convention, the dual basis, the space of row vectors as the dual of the space IK
n
of column
vectors, the matrix of an operator relative to a pair of bases
Oct. 3:
operators, linear equations, homogeneous linear equations, the image of
T
as the space
of those
y
for which the equation
Tx
=
y
is solvable, the kernel of
T
as the space of
solutions to the homogeneous linear equation
Tx
= 0, surjectivity and injectivity in terms
of existence and uniqueness of solutions, injectivity and linear independence, surjectivity
and the spanning property, the column space as the image of a matrix operator, linear
isomorphisms, isomorphisms between two spaces of the same ﬁnite dimension (which always
exist, and may be equivalently characterized as being just injective, or just surjective), the
Fredholm alternative
Oct. 6:
RANK VS. NULLITY (the dimension theorem), cosets of a vector subspace, the
set of solutions to a linear equation, solving linear equations by Gaussian elimination
Oct. 8: TEST 1
Oct. 10:
INVERSES of linear isomorphisms, reduced row echelon form, use of Gaus
sian elimination for ﬁnding the inverse of an invertible square matrix, diﬀerentiation vs.
integration, the natural isomorphism
δ
:
V
→
V
**
for a ﬁnitedimensional space
V
, the
B´
ezout theorem and its main consequence (a nonzero polynomial of degree
n
cannot have
more than
n
roots), the polynomial interpolation formula, the sum of two subsets in a
vector space, the case of two subspaces
Oct. 13:
QUOTIENT SPACES, addition of cosets, the quotient projection operator
P
:
V
→
V/W
, functions up to an additive constant, the dimension formula dim
V/W
=
dim
V

dim
W
, constructing a basis for
V/W
Oct. 15:
EIGENVECTORS and eigenvalues of an endomorphism, eigenspaces, invariant