\(% These macros are automatically generated from the "macros" % XML element. Make permanent edits there. % % History % 2004/01/01 Initiated for FCLA, evolved from there % 2006/09/17 Converted _, ^ to \sb, \sp for TeX4ht % 2014/02/01 Updated for MathBook XML projects % Obsolete in FCLA: \codeindent, \computerfont, \define % Change: MathJax wants \lt, so replaced by \lteval % 2014/02/22 New: \orderof, \reals, \per % %%%%%%%%%%%%%%%%%%%%% % % Conveniences % %%%%%%%%%%%%%%%%%%%%% % % Order of (asymptotically limit of fraction is 1) % Usage: \orderof{some function} % \newcommand{\orderof}[1]{\sim #1} % % Integers % Usage: \Z \newcommand{\Z}{\mathbb{Z}} % % Real numbers, as set of scalars % Usage: \reals \newcommand{\reals}{\mathbb{R}} % % n-space over real field % Usage: \complex{integer-dimension} \newcommand{\real}[1]{\mathbb{R}^{#1}} % % Complex numbers, as set of scalars % Usage: \complexes \newcommand{\complexes}{\mathbb{C}} % % n-space over complex field % Usage: \complex{integer-dimension} \newcommand{\complex}[1]{\mathbb{C}^{#1}} % % Complex conjugation (scalar, vector, matrix) % Usage: \conjugate{object} \newcommand{\conjugate}[1]{\overline{#1}} % % Complex number modulus % Usage: \modulus{a+bi} % Presumes math mode \newcommand{\modulus}[1]{\left\lvert#1\right\rvert} % % Zero vector % Usage: \zerovector \newcommand{\zerovector}{\vect{0}} % % Zero matrix % Usage: \zeromatrix, use a subscript when size is important \newcommand{\zeromatrix}{\mathcal{O}} % % Inner product (brackets, not quadratic form) % Usage: \innerproduct{a-vector}{a-vector} \newcommand{\innerproduct}[2]{\left\langle#1,\,#2\right\rangle} % % Norm of a vector % Usage: \norm{a-vector} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} % % Dimension % Usage: \dimension{vector-space-letter} \newcommand{\dimension}[1]{\dim\left(#1\right)} % % Nullity % Usage: \nullity{matrix-or-lintrans-letter} \newcommand{\nullity}[1]{n\left(#1\right)} % % Rank % Usage: \rank{matrix-or-lintrans-letter} \newcommand{\rank}[1]{r\left(#1\right)} % % Direct sum % Usage: \ds between a couple of subspaces % \newcommand{\ds}{\oplus} % % Determinant of a matrix (functional) % Usage: \detname{A} \newcommand{\detname}[1]{\det\left(#1\right)} % % Determinant of a matrix (vertical bars) % Usage: \detbars{A} \newcommand{\detbars}[1]{\left\lvert#1\right\rvert} % % Trace of a Matrix % Usage: \trace{matrix name} \newcommand{\trace}[1]{t\left(#1\right)} % % Square Root of a Matrix % Usage: \sr{a-matrix} \newcommand{\sr}[1]{#1^{1/2}} % %%%%%%%%%%%%%%%%%%%%% % % Subspace Constructions % %%%%%%%%%%%%%%%%%%%%% % % Span of a set of vectors % \span and \sp are used by TeX for other things % Usage: \spn{set-of-vectors} \newcommand{\spn}[1]{\left\langle#1\right\rangle} % % Null space of a matrix % Usage: \nsp{A} \newcommand{\nsp}[1]{\mathcal{N}\!\left(#1\right)} % % Column space of a matrix % Usage: \csp{A} \newcommand{\csp}[1]{\mathcal{C}\!\left(#1\right)} % % Row space of a matrix % Usage: \rsp{A} \newcommand{\rsp}[1]{\mathcal{R}\!\left(#1\right)} % % Left null space of a matrix % Usage: \lns{A} \newcommand{\lns}[1]{\mathcal{L}\!\left(#1\right)} % % Orthogonal complement of a vector space % Avoiding TeX's \perp % Usage: \per{A} \newcommand{\per}[1]{#1^\perp} % %%%%%%%%%%%%%%%%%%%%% % % Systems of Equations % %%%%%%%%%%%%%%%%%%%%% % % In-line form of an augmented matrix for a system of equations % Usage: \augmented{coefficient-matrix}{constant-vector} \newcommand{\augmented}[2]{\left\lbrack\left.#1\,\right\rvert\,#2\right\rbrack} % % Notation for a linear system before introducing matrix multiplication % Usage: \linearsystem{coefficient-matrix}{constant-vector} \newcommand{\linearsystem}[2]{\mathcal{LS}\!\left(#1,\,#2\right)} % % Notation for a homogenous system before introducing matrix multiplication % Usage: \homosystem{coefficient-matrix} \newcommand{\homosystem}[1]{\linearsystem{#1}{\zerovector}} % %%%%%%%%%%%%%%%%%%%%% % % Row Operations, Echelon Form % %%%%%%%%%%%%%%%%%%%%% % % Row operations on matrices % % Three commands to shorten up descriptions of gaussian elimination % % Usage: \rowopswap{row-i}{row-j} % Usage: \rowopmult{scalar}{row-i} % Usage: \rowopadd{scalar}{row-multiplied}{row-added-to} \newcommand{\rowopswap}[2]{R_{#1}\leftrightarrow R_{#2}} \newcommand{\rowopmult}[2]{#1R_{#2}} \newcommand{\rowopadd}[3]{#1R_{#2}+R_{#3}} % % Mark leading 1's in echelon form with fbox % Usage: \leading{a-1-usually} \newcommand{\leading}[1]{\fbox{#1}} % % Row-reduce arrow % Usage: \rref inbetween a matrix and its reduced row-echelon form \newcommand{\rref}{\xrightarrow{\text{RREF}}} % % Elementary Matrices % Usage: \elemswap{subscript}{subscript} % Usage: \elemmult{scalar}{subscript} % Usage: \elemadd{scalar}{subscript-mult}{subscript-target} % \newcommand{\elemswap}[2]{E_{#1,#2}} \newcommand{\elemmult}[2]{E_{#2}\left(#1\right)} \newcommand{\elemadd}[3]{E_{#2,#3}\left(#1\right)} % %%%%%%%%%%%%%%%%%%%%% % % 2-D Constructions (Lists, Vectors, Matrices) % %%%%%%%%%%%%%%%%%%%%% % % A list of scalars of generic length % Usage: \scalarlist{scalar letter}{terminal subscript} \newcommand{\scalarlist}[2]{{#1}_{1},\,{#1}_{2},\,{#1}_{3},\,\ldots,\,{#1}_{#2}} % % Vector styling, bold (or use wiggles, arrows, whatever) % Subscripts go outside this construction % Usage: \vect{a symbol to use as a vector} % Have to already be in math mode % \newcommand{\vect}[1]{\mathbf{#1}} % % A column vector % Usage: \colvector{list-delimited-by-\\} % \newcommand{\colvector}[1]{\begin{bmatrix}#1\end{bmatrix}} % % A generic vector with components % Usage: \vectorcomponents{component-letter}{final-subscript} \newcommand{\vectorcomponents}[2]{\colvector{#1_{1}\\#1_{2}\\#1_{3}\\\vdots\\#1_{#2}}} % % A list of vectors of generic length % Usage: \vectorlist{vector letter}{terminal subscript} \newcommand{\vectorlist}[2]{\vect{#1}_{1},\,\vect{#1}_{2},\,\vect{#1}_{3},\,\ldots,\,\vect{#1}_{#2}} % % Vector entries, entry i of vector v % (vector-expession still needs \vect, etc.) % Usage: \vectorentry{vector-expression}{single-subscript} \newcommand{\vectorentry}[2]{\left\lbrack#1\right\rbrack_{#2}} % % Matrix entries, entry i,j of matrix A % Usage: \matrixentry{matrix-expression}{paired-subscripts} % \newcommand{\matrixentry}[2]{\left\lbrack#1\right\rbrack_{#2}} % % A generic linear combination % Usage: \lincombo{scalar letter}{vector letter}{terminal subscript} \newcommand{\lincombo}[3]{#1_{1}\vect{#2}_{1}+#1_{2}\vect{#2}_{2}+#1_{3}\vect{#2}_{3}+\cdots +#1_{#3}\vect{#2}_{#3}} % % Matrix, column by column, as vectors % Usage: \matrixcolumns{matrix letter}{terminal subscript} \newcommand{\matrixcolumns}[2]{\left\lbrack\vect{#1}_{1}|\vect{#1}_{2}|\vect{#1}_{3}|\ldots|\vect{#1}_{#2}\right\rbrack} % %%%%%%%%%%%%%%%%%%%%% % % Special Matrices % %%%%%%%%%%%%%%%%%%%%% % % Transpose of a matrix % Usage: \transpose{A} \newcommand{\transpose}[1]{#1^{t}} % % Inverse of a matrix % Usage: \inverse{A} \newcommand{\inverse}[1]{#1^{-1}} % % Submatrix (for minors, determinants) % Usage: \submatrix{matrix-name}{delete-row}{delete-col} \newcommand{\submatrix}[3]{#1\left(#2|#3\right)} % % Adjoint of a matrix (twice) % This macro is a convenience to call \transpose and \conjugate properly % It shouldn't need to be modified (or mathematical meanings will change) % Usage: \adj{A} \newcommand{\adj}[1]{\transpose{\left(\conjugate{#1}\right)}} % % This macro controls the symbol used for the adjoint % It can be edited to taste % Usage: \adjoint{A} \newcommand{\adjoint}[1]{#1^\ast} % %%%%%%%%%%%%%%%%%%%%% % % Sets % %%%%%%%%%%%%%%%%%%%%% % % A convenience for simple sets % Usage: \set{list of element} \newcommand{\set}[1]{\left\{#1\right\}} % % Sets with vertical bar, "such that", sized for objects, not condition % Usage: \setparts{objects}{condition} % %%\newcommand{\setparts}[2]{\left\{ #1\mid#2\right\}} %%\newcommand{\setparts}[2]{\left\{\left. #1\right\rvert#2\right\}} \newcommand{\setparts}[2]{\left\lbrace#1\,\middle|\,#2\right\rbrace} % % Set Cardinality % Usage: \card{a-set-letter} \newcommand{\card}[1]{\left\lvert#1\right\rvert} % % Set Union % Use \cup % % Set Intersection % Use \cap % % Set Complement % Usage: \setcomplement{a-set-letter} \newcommand{\setcomplement}[1]{\overline{#1}} % %%%%%%%%%%%%%%%%%%%%% % % Eigenvalues and Eigenspaces % %%%%%%%%%%%%%%%%%%%%% % % Characteristic polynomial % Usage: \charpoly{matrix-letter}{variable-letter} \newcommand{\charpoly}[2]{p_{#1}\left(#2\right)} % % Eigenspace % Usage: \eigenspace{matrix-letter}{eigenvalue-letter} \newcommand{\eigenspace}[2]{\mathcal{E}_{#1}\left(#2\right)} % % 2013/10/03 Including ampersands is problematic here, % think about fixes later % 2014/02/22 Limited testing, seems & is fine for HTML and LaTeX % Eigensystem (presumes wrapped in an mrow within md) % Usage: \eigensystem{matrixletter}{eigenvalue}{list of basis vectors} \newcommand{\eigensystem}[3]{\lambda&=#2&\eigenspace{#1}{#2}&=\spn{\set{#3}}} % % Generalized Eigenspace % Usage: \geneigenspace{lin-trans-letter}{eigenvalue-letter} \newcommand{\geneigenspace}[2]{\mathcal{G}_{#1}\left(#2\right)} % % Algebraic multiplicty % Usage: \algmult{matrix-letter}{eigenvalue-letter} \newcommand{\algmult}[2]{\alpha_{#1}\left(#2\right)} % % Geometric multiplicty % Usage: \geomult{matrix-letter}{eigenvalue-letter} \newcommand{\geomult}[2]{\gamma_{#1}\left(#2\right)} % % Index (of eigenvalue) % Usage: \indx{matrix-letter}{eigenvalue-letter} \newcommand{\indx}[2]{\iota_{#1}\left(#2\right)} % %%%%%%%%%%%%%%%%%%%%% % % Linear Transformations % %%%%%%%%%%%%%%%%%%%%% % % MathJax defines \lt to ease XML confusion % % Linear transformation definition % Usage: \ltdefn{name-letter}{domain}{range} \newcommand{\ltdefn}[3]{#1\colon #2\rightarrow#3} % % Linear transformation evaluation % Usage: \lteval{name-letter}{input} % Replaces old \lt desired by MathJax \newcommand{\lteval}[2]{#1\left(#2\right)} % % Linear transformation inverse % Usage: \ltinverse{name-letter} \newcommand{\ltinverse}[1]{#1^{-1}} % % Linear transformation restriction % Usage: \restrict{name-letter}{subspace-letter} \newcommand{\restrict}[2]{{#1}|_{#2}} % % Linear transformation preimage % Usage: \preimage{name-letter}{codomain-element} \newcommand{\preimage}[2]{#1^{-1}\left(#2\right)} % % Range of a linear transformation % TeX uses \range for something else % Usage: \rng{T} \newcommand{\rng}[1]{\mathcal{R}\!\left(#1\right)} % % Kernel of a linear transformation % TeX uses \ker to do something different % Usage: \krn{T} \newcommand{\krn}[1]{\mathcal{K}\!\left(#1\right)} % % Linear transformation composition % Usage: \compose{function-name}{function-name} \newcommand{\compose}[2]{{#1}\circ{#2}} % % Vector space of linear transformations % Usage: \vslt{domains}{codomains} % Presumes math mode \newcommand{\vslt}[2]{\mathcal{LT}\left(#1,\,#2\right)} % %%%%%%%%%%%%%%%%%%%%% % % Vector and Matrix Representations % %%%%%%%%%%%%%%%%%%%%% % % Isomorphism symbol % Usage: \isomorphic \newcommand{\isomorphic}{\cong} % % Similarity % Usage: \similar{inner-matrix}{outer-invertible-matrix} % Rearranging this will not "fix" all desired changes throughout % \newcommand{\similar}[2]{\inverse{#2}#1#2} % % Vector representation function name % Usage: \vectrepname{basis-letter} \newcommand{\vectrepname}[1]{\rho_{#1}} % % Vector representation output % Usage: \vectrep{basis-letter}{input} \newcommand{\vectrep}[2]{\lteval{\vectrepname{#1}}{#2}} % % Vector representation inverse function name % (Added later, not used consistently in FCLA) % Usage: \vectrepinvname{basis-letter} \newcommand{\vectrepinvname}[1]{\ltinverse{\vectrepname{#1}}} % % Vector representation inverse output % Usage: \vectrepinv{basis-letter}{input} \newcommand{\vectrepinv}[2]{\lteval{\ltinverse{\vectrepname{#1}}}{#2}} % % Matrix representation % Usage: \matrixrep{transformation-letter}{domain-basis-letter}{codomain-basis-letter} \newcommand{\matrixrep}[3]{M^{#1}_{#2,#3}} % % Matrix representation column-by-colum % Usage: \matrixrepcolumns{transformation-letter}{codomain-basis-letter}{codomain-basis-vector-letter}{final-index} \newcommand{\matrixrepcolumns}[4]{\left\lbrack \left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{1}}}\right|\left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{2}}}\right|\left.\vectrep{#2}{\lteval{#1}{\vect{#3}_{3}}}\right|\ldots\left|\vectrep{#2}{\leval{#1}{\vect{#3}_{#4}}}\right.\right\rbrack} % % Change of basis matrix % Usage: \cbm{domain-basis-letter}{codomain-basis-letter} \newcommand{\cbm}[2]{C_{#1,#2}} % %%%%%%%%%%%%%%%%%%%%% % % Canonical Forms % %%%%%%%%%%%%%%%%%%%%% % % Jordan blocks % Usage: \jordan{size}{diagonal-element} \newcommand{\jordan}[2]{J_{#1}\left(#2\right)} % %%%%%%%%%%%%%%%%%%%%% % % Hadamard Matrices % Contributed by Elizabeth Million % %%%%%%%%%%%%%%%%%%%%% % % Hadamard Product % Usage: \hadamard{a-matrix}{a-matrix} \newcommand{\hadamard}[2]{#1\circ #2} % % Hadamard identity matrix % Usage: \hadamardidentity{paired-subscripts-size-of-matrix} \newcommand{\hadamardidentity}[1]{J_{#1}} % % Hadamard inverse matrix % Usage: \hadamardinverse{matrix-expression} \newcommand{\hadamardinverse}[1]{\widehat{#1}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & }\)

A Second Course in Linear Algebra

Robert A. Beezer

Department of Mathematics and Computer Science

University of Puget Sound

beezer@pugetsound.edu

DRAFT March 3, 2016 DRAFT