review1

# review1 - These lecture summaries may also be viewed online...

This preview shows pages 1–2. Sign up to view the full content.

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 , 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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/08/2010 for the course NE 112 taught by Professor Vanelli during the Spring '10 term at Waterloo.

### Page1 / 6

review1 - These lecture summaries may also be viewed online...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online