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Lecture Summaries for Linear Algebra M51A
\
x
" refers to the section in the textbook.
Lecture 01
:
Introduction
. An introduction to the course, listing the topics to be covered;
remarks concerning the rigor and formality with which the theory will be developed.
Lecture 02
(
x
1.1) :
Matrices
. De¯nitions, notations, and examples of: matrix, element,
order,
a
ij
, (as well as
A
,
A
i;j
,
A
(
i;j
),
i; j
-entry of
A
), [
a
], row index, column index, main (or
principal) diagonal, diagonal element, square matrix, row matrix, column matrix, components,
dimension,
n
-tuple, equality of matrices.
Lecture 03
(
x
1.1) :
Matrix Addition and Scalar Multiplication
. De¯nition and
example of matrix addition; theorem asserting commutativity and associativity of (matrix) ad-
dition; de¯nition of the zero matrix; theorem asserting that
0
is an additive identity element;
de¯nition of matrix subtraction; de¯nition and example of scalar multiplication; theorem assert-
ing the basic associative and distributive propertiesof scalar multiplication; some basic properties
(left as exercises) relating matrix addition, subtraction, and scalar multiplication.
Lecture 04
(
x
1.2) :
Matrix Multiplication
. De¯nition of matrix multiplication; exam-
ples showing that matrix multiplication is not commutative, that a product of two non-zero
matrices can return the zero matrix, that \cancellation" does not hold for matrix multiplication;
theorem asserting that matrix multiplication is associative and distributes over matrix addition.
Lecture 05
(
x
1.3) :
Transpose and Symmetry
. De¯nition, examples, and basic properties
of the transpose operation; notation
A
n
for
n >
0 and for
n
= 0; de¯nitions, examples, and basic
properties of symmetric, skew-symmetric, diagonal, and identity matrices.
Lecture 06
(
x
1.3) :
Partitions and Special Forms of Matrices
.
De¯nitions and
examples of submatrix, partitioned matrix, block, symmetrically partitioned matrix, diagonal
block of a symmetrically partitioned matrix, zero row of a matrix, nonzero row of a matrix,
row-reduced form, upper triangular, lower triangular; theorem asserting that the product of two
upper triangular matrices is upper triangular, and the product of two lower triangular matrices
is lower triangular.
Lecture 07
(
x
1.4) :
Linear Systems of Equations
. De¯nition and examples of a system
of
m
linear equations in
n
variables, and its solutions; matrix representation of such a system;
theorem asserting that if
x
1
and
x
2
are distinct solutions to
A
x
=
b
, then
®
x
1
+
¯
x
2
is also
a solution whenever
®
+
¯
= 1; corollary asserting that a system of linear equations has either
0, 1, or in¯nitely many solutions; geometrical perspective of this corollary in 2-space and 3-
space; de¯nitions of homogeneous, inhomogeneous, consistent, inconsistent, and trivial solution;
observation that the trivial solution makes every homogeneous system consistent.