Linear Algebra is the third book in the Life of Fred University Mathematics Series (if this series is broken down into University Math Set 1 and University Math Set 2, it is the first book of "University/College Set 2"), and covers junior-year algebra that should only be taken after completing calculus.
Four chapters are included that deal with the systems of linear equations: systems with exactly one solution, systems with many solutions, systems with no solution, and systems evolving over time. Four additional half-chapters build upon that foundation to delve into more abstract topics such as vector spaces and inner product spaces. "Your Turn to Play" sections are interspersed throughout each chapter and involve students in the process of learning through practice. After completing the problems, fully worked solutions are given for students to check their work. At the end of each chapter are six "city" problem sets (each named after a city). All "city" answers are provided in this expanded edition.
Life of Fred Linear Algebra covers: Solving systems of equations with one, many, and no solutions. Gauss-Jordan elimination. Gaussian elimination. Matrices. LU-decomposition. Vector spaces. Inner product spaces. Gram-Schmidt orthogonalization process. Fourier series. Data fitting. Linear Transformations. Linear functionals. Dual spaces. Eigenvalues and eigenvectors. Markov chains.
There are many different "algebras":
- Beginning Algebra = Algebra 1 = first year of high school algebra
- Advanced 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 $61. And the fun of reading about Fred's adventures.
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