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\(\require{cancel}\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 }} 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\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 \vdots\\ 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 \vdots\\ 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 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#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 \\ 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Appendix B Definitions

1.1 Sets and functions

Definition 1.1.1 Sets

Definition 1.1.2 Set equality

Definition 1.1.3 Set inclusion (subsets)

Definition 1.1.5 Set-builder notation

Definition 1.1.8 Basic set operations

Definition 1.1.9 Common mathematical sets

Definition 1.1.10 Cartesian product

Definition 1.1.13 Functions

Definition 1.1.17 Function equality

Definition 1.1.18 Image of a set

Definition 1.1.19 Injective, surjective, bijective

Definition 1.1.21 Function composition

Definition 1.1.22 Identity and inverse functions

1.2 Logic

Definition 1.2.1 Logical operators

Definition 1.2.5

2.1 Systems of linear equations

Definition 2.1.1 Linear equations

Definition 2.1.3 Systems of linear equations

Definition 2.1.5 Solutions to linear systems

Definition 2.1.8 Elementary operations on linear systems

Definition 2.1.11 Equivalence of linear systems

2.2 Gaussian elimination

Definition 2.2.1 Augmented matrix

Definition 2.2.3 Row echelon form

Definition 2.2.5 Elementary row operations on matrices

Definition 2.2.8 Gaussian elimination

Definition 2.2.9 Gauss-Jordan elimination

2.3 Solving linear systems

Definition 2.3.1 Free and leading variables

Definition 2.3.4 Consistent and inconsistent systems

3.1 Matrices and their arithmetic

Definition 3.1.2 Matrix

Definition 3.1.3

Definition 3.1.5 Matrix equality

Definition 3.1.7 Square matrices, row vectors, column vectors, zero matrices

Definition 3.1.9 Matrix addition and subtraction

Definition 3.1.11 Scalar multiplication of matrices

Definition 3.1.13 Linear combination of matrices

Definition 3.1.17 Matrix multiplication

Definition 3.1.24 Matrix transposition

3.2 Algebra of matrices

Definition 3.2.2 Additive inverse of a matrix

Definition 3.2.3 Identity matrix

3.3 Invertible matrices

Definition 3.3.1 Invertible matrix

Definition 3.3.9 Matrix powers

Definition 3.3.10

3.4 The invertibility theorem

Definition 3.4.1

Definition 3.4.7

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 3.5.14 Adjoint matrix

4.1 Real vector spaces

Definition 4.1.1 Vector space

Definition 4.1.3 Vector space of \(m\times n\) matrices

Definition 4.1.4 Vector space of real \(n\)-tuples

Definition 4.1.6 Zero vector space

Definition 4.1.7 The vector space of infinite real sequences

Definition 4.1.8 Real-valued functions

Definition 4.1.10 Vector space of positive real numbers

Definition 4.1.11 Linear combination of vectors

4.2 Linear transformations

Definition 4.2.1 Linear transformations

Definition 4.2.3 Zero and identity transformation

Definition 4.2.8 Matrix transformations

Definition 4.2.12 Rotation in the plane

Definition 4.2.16 Reflection through a line

4.3 Subspaces

Definition 4.3.1 Subspace

Definition 4.3.8

Definition 4.3.12 Null space and image

Definition 4.3.19 Symmetric and skew-symmetric matrices

4.4 Span and linear independence

Definition 4.4.1 Span

Definition 4.4.5 Spanning set

Definition 4.4.8 Linear independence

4.5 Bases and dimension

Definition 4.5.1 Basis

Definition 4.5.9 Cardinality of a set

Definition 4.5.12 Dimension of a vector space

4.6 Rank-nullity theorem and fundamental spaces

Definition 4.6.1 Rank and nullity

Definition 4.6.5 Fundamental spaces

4.7 Isomorphisms

Definition 4.7.6 Isomorphism

5.1 Inner product spaces

Definition 5.1.1 Inner product

Definition 5.1.13 Norm (or length) of a vector

Definition 5.1.15 Distance between vectors

Definition 5.1.19 Angle between vectors

5.2 Orthogonal bases and orthogonal projection

Definition 5.2.1 Orthogonal

Definition 5.2.3 Orthogonal and orthonormal bases

Definition 5.2.8 Orthogonal matrices

Definition 5.2.9 Orthogonal complement

6.1 Coordinate vectors and isomorphisms

Definition 6.1.1 Ordered bases

Definition 6.1.3 Coordinate vectors

6.2 Matrix representations of linear transformations

Definition 6.2.1

6.3 Change of basis

Definition 6.3.1 Change of basis matrix

Definition 6.3.6

6.4 Eigenvectors and eigenvalues

Definition 6.4.1

Definition 6.4.2

Definition 6.4.4

6.5 Diagonalization

Definition 6.5.1

Definition 6.5.9

Definition 6.5.16