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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Matrices and Linear Algebra">}
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\section{Matrices and Linear Algebra}

Linear algebra has its beginnings in the study and solution of systems of
linear equations. It is deeply rooted in analytic geometry and the theory of
systems of linear equations. Applications of linear algebra are found in
many fields, including economics, physics, sociology, and engineering.

\subsection{\ Matrices and Linear Algebra Contents}

\subsubsection{Systems of Linear Equations}

\hyperref{+ Introduction}{}{}{MA01_01.tex}

\hyperref{+ Solving Systems -- Back Substitution, Equivalent Systems}{}{}{%
MA01_02.tex}

\hyperref{+ Solving Systems -- Forward Elimination, Gaussian Elimination}{}{%
}{MA01_03.tex}

\hyperref{+ Vectors and Matrices}{}{}{MA01_04.tex}

\hyperref{+ Matrix Solution of Systems}{}{}{MA01_05.tex}

\hyperref{+ Matrix Multiplication}{}{}{MA01_06.tex}

\subsubsection{Vector Spaces}

\hyperref{+ Introduction}{}{}{MA02_01.tex}\textbf{\qquad }

\hyperref{+ Span}{}{}{MA02_02.tex}

\hyperref{+ Linear Independence}{}{}{MA02_03.tex}\textbf{\qquad }

\hyperref{+ Basis and Dimension}{}{}{MA02_04.tex}

\hyperref{+ Rank}{}{}{MA02_05.tex}

\hyperref{+ Linear Transformations}{}{}{MA02_06.tex}

\subsubsection{Orthogonality}

\hyperref{+ Perpendicular Vectors and Orthogonal Subspaces}{}{}{MA03_01.tex}

\hyperref{+ Projections onto Lines}{}{}{MA03_02.tex}

\hyperref{+ Projections and Least-Squares Approximations}{}{}{MA03_03.tex}

\hyperref{+ Orthogonal Matrices and Gram-Schmidt Orthonormalization}{}{}{%
MA03_04.tex}

\subsubsection{Determinants}

\hyperref{+ Properties}{}{}{MA04_01.tex}

\hyperref{+ Cross Product}{}{}{MA04_02.tex}

\hyperref{+ Adjoint Matrix}{}{}{MA04_03.tex}

\subsubsection{Eigenvalues and Eigenvectors}

\hyperref{+ Introduction}{}{}{MA05_01.tex}

\hyperref{+ Diagonalization}{}{}{MA05_02.tex}

\hyperref{+ Symmetric Matrices}{}{}{MA05_03.tex}

\hyperref{+ Complex Matrices}{}{}{MA05_04.tex}

\hyperref{+ Similar Matrices}{}{}{MA05_05.tex}

\subsection{\ Related Reference Topics\hfill}

\qquad {\small \hyperref{Algebra}{}{}{al_contents.tex}}

\subsection{\ Related Scientific Notebook Topics\hfill}

{\small \hyperref{\qquad Working With Matrices}{}{}{%
../../Help/DM8-1.tex#Matrix Algebra}}

{\small \hyperref{\qquad Standard Operations}{}{}{%
../../Help/DM8-2.tex#Standard matrix operations}}

{\small \hyperref{\qquad Row Operations and Echelon Forms}{}{}{%
../../Help/DM8-3.tex#Matrices, row operations}}

{\small \qquad \hyperref{Equations}{}{}{../../Help/DM8-4.tex#Systems of
equations}}

{\small \hyperref{\qquad Matrix Operators}{}{}{../../Help/DM8-5.tex#Trace}}

{\small \hyperref{\qquad Polynomials and Vectors Associated with a Matrix}{}{%
}{../../Help/DM8-6.tex#Characteristic, polynomial}}

{\small \hyperref{\qquad Vector Spaces Associated with a Matrix}{}{}{%
../../Help/DM8-7.tex#Row space}}

{\small \hyperref{\qquad Special Forms of Matrices}{}{}{%
../../Help/DM8-8.tex#Smith normal form}}

{\small \hyperref{\qquad Sample Problems}{}{}{../../Help/DM8-9.tex#Exercises}%
}

\rule{0.67in}{0.01in}

\quad {\small Computing Techniques Index entries: \hyperref{Echelon forms}{}{%
}{../../Help/DMIndex.tex#Echelon forms}, \hyperref{Eigenvectors}{}{}{%
../../Help/DMIndex.tex#Eigenvalues and eigenvectors}, \hyperref{Fill matrix}{%
}{}{../../Help/DMIndex.tex#Fill matrix}, \hyperref{Gaussian elimination}{}{}{%
../../Help/DMIndex.tex#Gaussian elimination}, \hyperref{Gram-Schmidt
orthogonalization}{}{}{../../Help/DMIndex.tex#Gram-Schmidt orthogonalization}%
, \hyperref{Jordan form}{}{}{../../Help/dmindex.tex#Jordan form}, \hyperref{%
Matrices}{}{}{../../Help/dmindex.tex#Matrices}, \hyperref{Matrix operators}{%
}{}{..\Help\dmindex.tex#Matrix operators}, \hyperref{Orthogonal matrices}{}{%
}{..\Help\dmindex.tex#Orthogonal matrices}, \ \hyperref{Orthonormal bases}{}{%
}{..\Help\dmindex.tex#Orthonormal bases}, \hyperref{Positive definite
symmetric matrices}{}{}{..\Help\dmindex.tex#Positive definite symmetric
matrices}, \hyperref{QR factorization}{}{}{..\Help\dmindex.tex#QR
factorization}, \hyperref{row basis}{}{}{..\Help\dmindex.tex#Row basis}, 
\hyperref{SVD}{}{}{..\Help\dmindex.tex#SVD}, \hyperref{Symmetric matrices}{}{%
}{..\Help\dmindex.tex#Symmetric matrices}, \hyperref{Transpose}{}{}{%
..\Help\dmindex.tex#Transpose}, \hyperref{Vector}{}{}{..\Help%
\dmindex.tex#Vector}}

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