Definitions

Skip to main content

\(\newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\T}{{\mathbb T}} \newcommand{\F}{{\mathbb F}} \newcommand{\HH}{{\mathbb H}} \newcommand{\compose}{\circ} \newcommand{\bolda}{{\mathbf a}} \newcommand{\boldb}{{\mathbf b}} \newcommand{\boldc}{{\mathbf c}} \newcommand{\boldd}{{\mathbf d}} \newcommand{\bolde}{{\mathbf e}} \newcommand{\boldi}{{\mathbf i}} \newcommand{\boldj}{{\mathbf j}} \newcommand{\boldk}{{\mathbf k}} \newcommand{\boldn}{{\mathbf n}} \newcommand{\boldp}{{\mathbf p}} \newcommand{\boldq}{{\mathbf q}} \newcommand{\boldr}{{\mathbf r}} \newcommand{\bolds}{{\mathbf s}} \newcommand{\boldt}{{\mathbf t}} \newcommand{\boldu}{{\mathbf u}} \newcommand{\boldv}{{\mathbf v}} \newcommand{\boldw}{{\mathbf w}} \newcommand{\boldx}{{\mathbf x}} \newcommand{\boldy}{{\mathbf y}} \newcommand{\boldz}{{\mathbf z}} \newcommand{\boldzero}{{\mathbf 0}} \newcommand{\boldmod}{\boldsymbol{ \bmod }} \newcommand{\boldT}{{\mathbf T}} \newcommand{\boldN}{{\mathbf N}} \newcommand{\boldB}{{\mathbf B}} \newcommand{\boldF}{{\mathbf F}} \newcommand{\boldS}{{\mathbf S}} \newcommand{\boldG}{{\mathbf G}} \newcommand{\boldK}{{\mathbf K}} \newcommand{\boldL}{{\mathbf L}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\NS}{null} \DeclareMathOperator{\RS}{row} \DeclareMathOperator{\CS}{col} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\nullity}{nullity} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Aff}{Aff} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\mdeg}{mdeg} \DeclareMathOperator{\Lt}{Lt} \DeclareMathOperator{\Lc}{Lc} \DeclareMathOperator{\disc}{disc} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\curl}{curl} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\diver}{div} \DeclareMathOperator{\flux}{flux} \def\Gal{\operatorname{Gal}} \def\ord{\operatorname{ord}} \def\ML{\operatorname{M}} \def\GL{\operatorname{GL}} \def\PGL{\operatorname{PGL}} \def\SL{\operatorname{SL}} \def\PSL{\operatorname{PSL}} \def\GSp{\operatorname{GSp}} \def\PGSp{\operatorname{PGSp}} \def\Sp{\operatorname{Sp}} \def\PSp{\operatorname{PSp}} \def\Aut{\operatorname{Aut}} \def\Inn{\operatorname{Inn}} \def\Hom{\operatorname{Hom}} \def\End{\operatorname{End}} \def\ch{\operatorname{char}} \def\Zp{\Z/p\Z} \def\Zm{\Z/m\Z} \def\Zn{\Z/n\Z} \def\Fp{\F_p} \newcommand{\surjects}{\twoheadrightarrow} \newcommand{\injects}{\hookrightarrow} \newcommand{\bijects}{\leftrightarrow} \newcommand{\isomto}{\overset{\sim}{\rightarrow}} \newcommand{\floor}[1]{\lfloor#1\rfloor} \newcommand{\ceiling}[1]{\left\lceil#1\right\rceil} \newcommand{\mclass}[2][m]{[#2]_{#1}} \newcommand{\val}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\abs}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\valuation}[2][]{\left\lvert #2\right\rvert_{#1}} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\anpoly}{a_nx^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0} \newcommand{\anmonic}{x^n+a_{n-1}x^{n-1}\cdots +a_1x+a_0} \newcommand{\bmpoly}{b_mx^m+b_{m-1}x^{m-1}\cdots +b_1x+b_0} \newcommand{\bnpoly}{b_nx^n+b_{n-1}x^{n-1}\cdots +b_1x+b_0} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}} \newcommand{\normalin}{\trianglelefteq} \newcommand{\angvec}[1]{\langle #1\rangle} \newcommand{\varpoly}[2]{#1_{#2}x^{#2}+#1_{#2-1}x^{#2-1}\cdots +#1_1x+#1_0} \newcommand{\varpower}[1][a]{#1_0+#1_1x+#1_1x^2+\cdots} \newcommand{\limasto}[2][x]{\lim_{#1\rightarrow #2}} \def\ntoinfty{\lim_{n\rightarrow\infty}} \def\xtoinfty{\lim_{x\rightarrow\infty}} \def\ii{\item} \def\bb{\begin{enumerate}} \def\ee{\end{enumerate}} \def\ds{\displaystyle} \def\p{\partial} \newcommand{\abcdmatrix}[4]{\begin{bmatrix}#1\amp #2\\ #3\amp #4 \end{bmatrix} } \newenvironment{amatrix}[1][ccc|c]{\left[\begin{array}{#1}}{\end{array}\right]} \newenvironment{linsys}[2][m]{ \begin{array}[#1]{@{}*{#2}{rc}r@{}} }{ \end{array}} \newcommand{\eqsys}{\begin{array}{rcrcrcr} a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\ a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m \end{array} } \newcommand{\numeqsys}{\begin{array}{rrcrcrcr} e_1:\amp a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp b_1\\ e_2: \amp a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp b_2\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ e_m: \amp a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp b_m \end{array} } \newcommand{\homsys}{\begin{array}{rcrcrcr} a_{11}x_{1}\amp +\amp a_{12}x_{2}\amp +\cdots+\amp a_{1n}x_{n}\amp =\amp 0\\ a_{21}x_{1}\amp +\amp a_{22}x_{2}\amp +\cdots+\amp a_{2n}x_{n}\amp =\amp 0\\ \amp \vdots\amp \amp \vdots \amp \amp \vdots \amp \\ a_{m1}x_{1}\amp +\amp a_{m2}x_{2}\amp +\cdots +\amp a_{mn}x_{n}\amp =\amp 0 \end{array} } \newcommand{\vareqsys}[4]{ \begin{array}{ccccccc} #3_{11}x_{1}\amp +\amp #3_{12}x_{2}\amp +\cdots+\amp #3_{1#2}x_{#2}\amp =\amp #4_1\\ #3_{21}x_{1}\amp +\amp #3_{22}x_{2}\amp +\cdots+\amp #3_{2#2}x_{#2}\amp =\amp #4_2\\ \vdots \amp \amp \vdots \amp \amp \vdots \amp =\amp \\ #3_{#1 1}x_{1}\amp +\amp #3_{#1 2}x_{2}\amp +\cdots +\amp #3_{#1 #2}x_{#2}\amp =\amp #4_{#1} \end{array} } \newcommand{\genmatrix}[1][a]{ \begin{bmatrix} #1_{11} \amp #1_{12} \amp \cdots \amp #1_{1n} \\ #1_{21} \amp #1_{22} \amp \cdots \amp #1_{2n} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ #1_{m1} \amp #1_{m2} \amp \cdots \amp #1_{mn} \end{bmatrix} } \newcommand{\varmatrix}[3]{ \begin{bmatrix} #3_{11} \amp #3_{12} \amp \cdots \amp #3_{1#2} \\ #3_{21} \amp #3_{22} \amp \cdots \amp #3_{2#2} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ #3_{#1 1} \amp #3_{#1 2} \amp \cdots \amp #3_{#1 #2} \end{bmatrix} } \newcommand{\augmatrix}{ \begin{amatrix}[cccc|c] a_{11} \amp a_{12} \amp \cdots \amp a_{1n} \amp b_{1}\\ a_{21} \amp a_{22} \amp \cdots \amp a_{2n} \amp b_{2}\\ \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots\\ a_{m1} \amp a_{m2} \amp \cdots \amp a_{mn}\amp b_{m} \end{amatrix} } \newcommand{\varaugmatrix}[4]{ \begin{amatrix}[cccc|c] #3_{11} \amp #3_{12} \amp \cdots \amp #3_{1#2} \amp #4_{1}\\ #3_{21} \amp #3_{22} \amp \cdots \amp #3_{2#2} \amp #4_{2}\\ \vdots \amp \vdots \amp \ddots \amp \vdots \amp \vdots\\ #3_{#1 1} \amp #3_{#1 2} \amp \cdots \amp #3_{#1 #2}\amp #4_{#1} \end{amatrix} } \newcommand{\spaceforemptycolumn}{\makebox[\wd\boxofmathplus]{\ }} \newcommand{\generalmatrix}[3]{ \left( \begin{array}{cccc} #1_{1,1} \amp #1_{1,2} \amp \ldots \amp #1_{1,#2} \\ #1_{2,1} \amp #1_{2,2} \amp \ldots \amp #1_{2,#2} \\ \amp \vdots \\ #1_{#3,1} \amp #1_{#3,2} \amp \ldots \amp #1_{#3,#2} \end{array} \right) } \newcommand{\colvec}[2][c]{\begin{amatrix}[#1] #2 \end{amatrix}} \newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}} \DeclareMathOperator{\size}{size} \DeclareMathOperator{\adjoint}{adj} \DeclareMathOperator{\sgn}{sgn} \newcommand{\restrictionmap}[2]{{#1}\mathpunct\upharpoonright\hbox{}_{#2}} \renewcommand{\emptyset}{\varnothing} \newcommand{\proj}[2]{\mbox{proj}_{#2}({#1}) } \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Appendix C Definitions

0.1 Sets

Definition 0.1.1 Sets

Definition 0.1.2 Set equality

Definition 0.1.3 Set inclusion (subsets)

Definition 0.1.5 Set-builder notation

Definition 0.1.8 Union, intersection, difference, and complement

Definition 0.1.9 Common mathematical sets

0.2 Functions

Definition 0.2.1 Functions

Definition 0.2.5 Function equality

Definition 0.2.6 Image of a set

Definition 0.2.7 Injective, surjective, bijective

Definition 0.2.9 Function composition

Definition 0.2.10 Identity and inverse functions

0.3 Tuples and Cartesian products

Definition 0.3.1 \(n\)-tuple

Definition 0.3.4 Cartesian product (finite)

Definition 0.3.5 I-tuple

Definition 0.3.6 Cartesian product (arbitrary)

0.4 Logic

Definition 0.4.1 Logical operators

Definition 0.4.5 Logical quantifiers

0.6 Complex numbers

Definition 0.6.1 Complex numbers

Definition 0.6.4 Complex addition and multiplication

Definition 0.6.9 Absolute value and complex conjugation

0.7 Polynomials

Definition 0.7.1 Polynomials

Definition 0.7.5 Degree of a polynomial

1.1 Vector space structure of \(\R^n\)

Definition 1.1.1 Real \(n\)-space

Definition 1.1.2 Vector operations of \(\R^n\)

Definition 1.1.4 Vector space

Definition 1.1.8 Vector space terminology for \(\R^n\)

Definition 1.1.10 Zero space

Definition 1.1.13 Linear combination

Definition 1.1.16 Vector difference

1.2 Inner product structure of \(\R^n\)

Definition 1.2.1 Dot product

Definition 1.2.3 Inner product

Definition 1.2.4 Weighted dot product

2.1 Systems of linear equations

Definition 2.1.1 Linear equations

Definition 2.1.3 Solutions to linear equations

Definition 2.1.4 Hyperplane

Definition 2.1.9 Systems of linear equations

Definition 2.1.14 Elementary operations on linear systems

Definition 2.1.15 Row equivalent linear systems

2.2 Gaussian elimination

Definition 2.2.1 Augmented matrix

Definition 2.2.5 Row echelon form

Definition 2.2.8 Elementary row operations on matrices

2.3 Solving linear systems

Definition 2.3.1 Free and leading variables

Definition 2.3.5 Consistent and inconsistent systems

3.1 Matrix arithmetic

Definition 3.1.2 Matrix

Definition 3.1.3 Matrix notation

Definition 3.1.5 Rows and columns of a matrix

Definition 3.1.6 Matrix equality

Definition 3.1.8 Matrices of particular shape

Definition 3.1.11 Matrix addition and scalar multiplication

Definition 3.1.13 Zero matrices

Definition 3.1.14 Additive inverse matrix

Definition 3.1.21 Matrix multiplication

Definition 3.1.33 Matrix transposition

3.2 Matrix algebra

Definition 3.2.4 Identity matrix

3.3 Invertible matrices

Definition 3.3.1 Invertible matrix

Definition 3.3.10 Matrix powers

Definition 3.3.11 Matrix polynomials

3.4 The invertibility theorem

Definition 3.4.1 Elementary matrices

Definition 3.4.7 Diagonal and triangular matrices

3.5 The determinant

Definition 3.5.1 Submatrix notation

Definition 3.5.3 The determinant

Definition 3.5.7 Minors and expansions along rows/columns

Definition 3.5.15 Adjoint matrix

4.1 Subspaces

Definition 4.1.1 Subspace

Definition 4.1.10 Null space of matrix

Definition 4.1.18 Trace of a matrix

Definition 4.1.19 Trace-zero, symmetric, and skew-symmetric

4.2 Span and linear independence

Definition 4.2.1 Span

Definition 4.2.7 Spanning set

Definition 4.2.10 Linear independence

4.3 Bases

Definition 4.3.1 Basis

4.4 Dimension

Definition 4.4.1 Cardinality of a set

Definition 4.4.2 Dimension of a vector space

4.5 Fundamental spaces

Definition 4.5.1 Fundamental spaces

Definition 4.5.12 Rank and nullity of matrix

5.1 Linear transformations

Definition 5.1.1 Linear transformations

Definition 5.1.8 Zero, identity, and scaling transformations

Definition 5.1.15 Matrix conjugation

Definition 5.1.25 Matrix transformations

Definition 5.1.28 Standard matrix of linear \(T\colon \R^n\rightarrow \R^m\)

Definition 5.1.33 Rotation in the plane

Definition 5.1.37 Reflection through a line

5.2 Null space, image, and isomophisms

Definition 5.2.1 Null space and image

Definition 5.2.8 Rank and nullity

Definition 5.2.15 Isomorphism

5.3 Coordinate vectors

Definition 5.3.1 Ordered bases

Definition 5.3.3 Coordinate vectors

5.4 Matrix representations of linear transformations

Definition 5.4.1 Matrix representations of linear transformations

5.5 Change of basis

Definition 5.5.1 Change of basis matrix

Definition 5.5.22 Similar matrices

6.1 Eigenvectors and eigenvalues

Definition 6.1.3 Eigenvectors and eigenvalues

Definition 6.1.15 Eigenspaces

Definition 6.1.18 Characteristic polynomial of a matrix

6.2 Diagonalization

Definition 6.2.1 Diagonalizable

Definition 6.2.3 Eigenbasis

Definition 6.2.26 Characteristic polynomial of a transformation

Definition 6.2.28 Algebraic/geometric multiplicity

7.1 Inner product spaces

Definition 7.1.1 Norm (or length) of a vector

Definition 7.1.6 Distance between vectors

Definition 7.1.11 Angle between vectors

7.2 Orthogonal bases

Definition 7.2.1 Orthogonality

Definition 7.2.5 Orthogonal and orthonormal bases

Definition 7.2.13 Orthogonal matrices

7.3 Orthogonal projection

Definition 7.3.1 Orthogonal complement

Definition 7.3.11 Projection onto a line

Definition 7.3.15 Orthogonal projection onto a plane

7.4 The spectral theorem

Definition 7.4.1 Self-adjoint operators

Definition 7.4.9 Orthogonally diagonalizable