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Lecture Summaries for Linear Algebra M51A
" refers to the section in the textbook.
. An introduction to the course, listing the topics to be covered;
remarks concerning the rigor and formality with which the theory will be developed.
. De¯nitions, notations, and examples of: matrix, element,
, (as well as
], row index, column index, main (or
principal) diagonal, diagonal element, square matrix, row matrix, column matrix, components,
-tuple, equality of matrices.
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
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.
. 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.
Transpose and Symmetry
. De¯nition, examples, and basic properties
of the transpose operation; notation
0 and for
= 0; de¯nitions, examples, and basic
properties of symmetric, skew-symmetric, diagonal, and identity matrices.
Partitions and Special Forms of Matrices
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.
Linear Systems of Equations
. De¯nition and examples of a system
linear equations in
variables, and its solutions; matrix representation of such a system;
theorem asserting that if
are distinct solutions to
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.