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These lecture summaries may also be viewed online by clicking the \L" icon at the top right of any lecture screen. 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 ij , A i;j , A ( i;j ), i;j -entry of A ), [ a ij ], 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 properties of 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.
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