5. Matrices and Transformations

5.1

Definition of a matrix. The terms element, row, column and dimension.
Introductory examples.

Elementary Matrices, included in Linear Algebra WebNotes, contains information on matrices, together with proofs and examples. See the index. Access to some examples is forbidden.

Also see Matrices, from Oregon State University's Web Study Guide for Vector Calculus.

5.2

Algebra of matrices. Equality, addition, subtraction; multiplication of two matrices.
The identity matrix.

See S.O.S. Mathematics - Matrix Algebra.

5.3

Determinants of matrices, and the condition for singularity of a matrix.

Determinants are discussed in Linear Algebra WebNotes. See the index.

Also see Determinants, from OSU's Web Study Guide for Vector Calculus.

5.4

The inverse of an square matrix.
The inverse of a composite,  (AB)^-1   =   B^-1A^-1.

5.5

Linear transformations of vectors in two dimensions and their matrix representation:
rotations; reflections and enlargements.
The geometric significance of a determinant

5.6

Composition of linear transformations   P, Q

5.7

Solution of linear equations (at most three equations in three unknowns).
Conditions for the existence of
(i) a unique solution;
(ii) no solution;
(iii) an infinity of solutions.

See Using Matrices to Solve Systems with Three or More Unknowns, an on-line tutorial accompanying the book Finite Mathematics & Calculus Applied to the Real World, by Stefan Waner and Steven Costenoble of Hofstra University. The three cases for solutions are explicitly discussed at the end of this tutorial. It is also possible to download a pivoting program for the TI-82 and TI-83 calculators.

See Systems of linear equations, from Linear Algebra WebNotes.
These notes were written by Mark Sapir, of the University of Nebraska.

Functions from linear algebra can be calculated at Java Script Linear Algebra.

Also see Gaussian Elimination, from Oregon State University's Web Study Guide for Vector Calculus. In this method, row operations are carried out on matrices.