Practice Makes Perfect Linear Algebra

Preface

1 Systems of linear equations and matrices

Systems of linear equations

General systems of linear equations

Matrices

Row transformations and equivalence of matrices

Row-echelon form

Homogeneous systems

2 Matrix algebra

Matrix arithmetic

Inverse of a square matrix

Properties of invertible matrices

Matrix solutions of systems of linear equations

Transpose of a matrix

3 Graphing calculators and matrices

Matrix menu

Inputting and editing a matrix

Matrix arithmetic

Calculating determinants

Transpose of a matrix

Solving linear systems using Gauss-Jordan elimination

Solving linear systems using X = A–1C

4 Special types of square matrices

Nonsingular matrices

Triangular, diagonal, and scalar matrices

Involutory, idempotent, and nilpotent matrices

Symmetric and skew-symmetric matrices

Orthogonal matrices

Hermitian and skew-Hermitian matrices

5 Determinants

Determinant of a square matrix

Cramer’s rule

Properties of determinant

6 Vectors in Rn

Vectors in two dimensions

Dot product of vectors

Vectors in Rn

Vectors as matrices

7 Vector spaces

Definitions and terminology of vector spaces

Linear independence

Basis

Dimension

Row space, column space, and null space

Rank and nullity

8 Inner product spaces

Definition and terminology for inner product spaces

Norm of a vector in an inner product space

Cauchy-Schwarz inequality and properties of the norm

Orthogonality in inner product spaces

Gram-Schmidt procedure

9 Linear transformations

Definition and terminology for linear transformations

Kernel and image of a linear transformation

Matrix representations of linear transformations

Change of basis

Algebra of linear transformations

Linear operators on R 2 and R 3

10 Eigenvalues and eigenvectors

The eigenvalue problem

Useful properties of eigenvalues

Diagonalization

Answer key