There are many different "algebras":

- Beginning Algebra = Algebra 1 = first year of high school algebra
- Advance Algebra = Algebra 2 = second year of high school algebra
- Linear Algebra (which is what this book is about)
- Modern Algebra = Abstract Algebra which is usually studied as an upper-division course at the university. It deals with groups, rings, fields, . . . (whatever those are!)
- Boolean Algebra
- and many more algebras

*Life of Fred: Linear Algebra *is usually studied after the two years of college calculus. You will study:

Systems of equations with lots of ways to solve them

All kinds of spaces: Vector, Inner Product, and Dual Spaces

Linear Transformations including linear functionals.

Here's a more complete description:

Chapter 1 Systems of Equations with One Solution

high school algebra, three equations with three unknowns

coefficient and augmented matrices

elementary row operations

Gauss-Jordan elimination

Gaussian elimination

Chapter 1½ Matrices

matrix addition A + B

scalar multiplication rA

matrix multiplication AB

matrix inverse A–1

proof of associative law of matrix multiplication (AB)C = A(BC)

elementary matrices

LU-decomposition

permutation matrices

Chapter 2 Systems of Equations with Many Solutions

four difficulties with Gauss-Jordan elimination

#1: a zero on the diagonal

#2: zeros “looking south”

#3: zeros “looking east”

#4: a row with all zeros except for the last column

free variables

echelon and reduced row-echelon matrices

general solutions

homogeneous systems

rank of a matrix

Chapter 2½ Vector Spaces

four properties of vector addition

a very short course in abstract algebra

four properties of scalar multiplication

five vector spaces

linear combinations and spanning sets

linear dependence/independence

basis for a vector space

coordinates with respect to a basis

dimension of a vector space

subspace of a vector space

row space, column space, null space, and nullity

Chapter 2¾ Inner Product Spaces

dot product

inner product

positive-definiteness

length of a vector (norm of a vector)

angle between two vectors

perpendicular vectors (orthogonality)

Gram-Schmidt orthogonalization process

orthonormal sets

Fourier series

harmonic analysis

double Fourier series

complex vector spaces with an inner product

orthogonal complements

Chapter 3 Systems of Equations with No Solution

overdetermined/underdetermined systems

discrete/continuous variables

the normal equation/“the best possible answer”

least squares solution

data fitting

model functions

Chapter 3½ Linear Transformations

rotation, reflection, dilation, projection, derivatives, matrix multiplication

linear transformations, linear mappings, vector space homomorphisms

linear operators

ordered bases

zero transformation, identity transformation

the equivalence of linear transformations and matrix multiplication

Hom(V , W )

linear functionals

dual spaces

second dual of V

Chapter 4 Systems of Equations into the Future

transition matrix

determinants

characteristic polynomial/characteristic equation

eigenvalues

algebraic multiplicity/geometric multiplicity

computation of A100

stochastic matrices

Markov chains

steady state vectors

regular matrices

absorbing states

similar matrices

systems of linear differential equations

Fibonacci numbers

computer programs for linear algebra

Index

Six problems sets at the end of each chapter.

A full course in linear algebra for $49. And the fun of reading about Fred's adventures.

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