**Course summary**: We study $\mathbb{R}^{n}$ and generalise it with the definitions of a real vector space and a Euclidean space (a finite dimensional real vector space with an inner product). We study real matrices, which form an infinite dimensional real vector space, and their applications to linear system of equations. ---- # 1. Real Vectors ### 1.1 Column Vectors in $\mathbb{R}^{n}$ | Definitions | Theorems | Examples | | ------------------------------------------------- | --------------------------------------------------------------------------------------- | ------------------------------------- | | [[Real n-Space]]. | [[Real n-Space forms Abelian Group]]. | | | [[Addition in Real n-Space]]. | [[Commutativity of Addition in Real n-Space]]. | | | [[Scalar Multiplication in Real n-Space]]. | [[Distributivity of Scalar Multiplication Over Addition in Real n-Space]]. | | | | [[Associativity of Scalar Multiplication in Real n-Space]]. | | | | [[Scalar Multiplication by Zero in Real n-Space]]; [[Additive Inverse of Real Vector]]; | | | [[Collinearity in Real n-Space]]. | | | | [[Linear Combination of Subset of Real n-Space]]. | | | | [[Standard basis of real n-space]]. | | | | [[Dot Product in Real n-Space]]. | [[Dot Product is an Inner Product on Real n-Space]]. | | | | | | | [[Euclidean Norm]]. | [[Non-negative Definiteness of Length of Real Vector]]. | | | | [[Length of Scaled Real Vector]]. | | | | [[Unit Vector in Direction of Non-Zero Real Vector]]. | | | [[Angle Between Nonzero Real Vectors]]. | [[Cauchy-Schwartz inequality]]. | | | | [[Euclidean spaces are normed spaces]]. | | | [[Kronecker Delta Function]]. | | | | [[Orthonormal Set of Real Vectors]]. | | | | | | | | | | [[Lines and planes in Real 3-Space]]. | ### 1.2 Linear Systems and Matrices | Definitions | Theorems | Examples | | ----------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------ | --------------------------------------------- | | [[Linear System of Equations]]. | | | | [[Solution to Linear System of Equations]]. | | | | [[Augmented Matrix of Linear System of Equations]]. | | | | [[Elementary Row Operations]]. | | | | [[Matrix]]. The set of all real [[Real Matrices\|matrices with real entries]] is denoted $\text{Mat}_{mn}(\mathbb{R}).$ | | | | [[Dual of linear map]]. | | | | [[Real Matrix Addition]]. | | | | [[Real Matrix Scalar Multiplication]]. | [[Distributivity of Scalar Multiplication over Addition of Real Matrices]]. | | | | *[[Real Matrices Form Real Vector Space]]. | | | [[Real Matrix Product]]. | | [[Matrix Multiplication is not Commutative]]. | | [[Real Identity Matrix]]. | [[Product with Real Identity Matrix]]; [[Product with Real Zero Matrix]]. | | | | [[Associativity of Multiplication of Real Matrices]]. | | | | [[Distributivity of Multiplication over Addition of Real Matrices]]. | | | [[Integer Power of Real Square Matrix]]. | | | | [[Inverse of Real Square Matrix]]. A square matrix whose inverse exists is called **invertible** and called **singular** otherwise. | [[Uniqueness of Inverse of Invertible Real Matrix]]. | [[Inverse of 2 by 2 Real Matrix]]. | | | [[Inverse of Product of Invertible Real Matrices]]. | | | | [[Inverse of Inverse of Invertible Real Matrix]]. | | | | | | | [[Elementary Matrices]]. | *[[Elementary Row Operation is Equivalent to Pre-Multiplying by Elementary Matrix]]. | | | | [[Elementary Matrices are Invertible]]. | | | | [[Elementary Row Operations do not Alter Set of Solutions to Linear System of Equations]]. | | | [[Reduced Row Echelon Form for Real Matrix]]. | *[[Existence of Reduced Row Echelon Form of Real Matrix]]. | | | | *[[Solution to Linear System of Equations in Reduced Row Echelon Form]]. | | | | *[[Infinitely Many Solutions to Homogeneous Linear System of Equations with More Variables Than Equations]]. | | | | | | | | *[[Matrix Inverse by Row Reduction]]. | | ### 1.3 Subspaces & Bases of $\mathbb{R}^{n}$ | Definitions | Theorems | Examples | | ------------------------------------------------ | ---------------------------------------------------------------------------------------- | -------- | | [[Subspace of Real n-Space]]. | | | | [[Span of Subset of Real n-Space]]. | [[Span is Subspace of Real n-Space]]. | | | [[Column Span of Real Matrix]]. | | | | [[Spanning Set of Real n-Space]]. | *[[Spanning Set of Real n-Space contains at Least n Elements]]. | | | [[Linearly Independent Subset of Real n-Space]]. | [[Two Elements of Real n-Space are Linearly Dependent iff Collinear]]. | | | | *[[Linearly Independent Subset of Real n-Space contains at Most n Elements]]. | | | [[Basis of Real n-Space]]. | [[Basis of Real n-Space has n Elements]]. | | | | [[Uniqueness of Expression of Element of Real n-Space as Linear Combination of Basis]]. | | | | *[[Matrix representations of linear map]]. | | | | | | | [[Elementary Column Operations]] | *[[Elementary Column Operation is Equivalent to Post-Multiplying by Elementary Matrix]]. | | | [[Reduced Column Echelon Form for Real Matrix]]. | [[Basis of Column Span of Real Matrix from Reduced Column Echelon Form]]. | | | [[Smith Normal Form for Real Matrix]]. | *[[Existence of Smith Normal Form of Real Matrix]]. | | # 2. Real Vector Spaces ### 2.1 Vector spaces | Definitions | Lemmas | Main Theorems | Examples | | ---------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------- | -------- | | [[Real Vector Space]]. | [[Scalar Multiple of Zero of Real Vector Space]]. | | | | | [[Scalar Multiplication by Zero in Real Vector Space]]. | | | | | [[Scalar Multiplication by -1 gives Additive Inverse in Real Vector Space]]. | | | | | | | | | [[Span of Subset of Real Vector Space]]. | | [[Span is Subspace of Real Vector Space]]. | | | [[Subspace of Real Vector Space]]. | | [[Intersection of Subspaces is Subspace of Real Vector Space]]. | | | [[Dimension of Sum of Finite Dimensional Vector Subspaces (Dimension Formula)]]. | | | | | | | | | | [[Finite Dimensional Real Vector Space]]. | | | | | [[Linearly Independent Subset of Real Vector Space]]. | [[Span of Linearly Independent Subset of Real Vector Space is Strongly Minimal]]. | [[Linearly Independent Subset of Finite Dimensional Real Vector Space is Finite]]. | | | | | [[Infinite Dimensional Real Vector Space contains Infinite Linearly Independent Subset]]. | | | [[Basis of a Real Vector Space]]. | | [[Basis of Real Vector Space iff Uniqueness of Expression of Vectors as Linear Combination of Elements]]. | | | | | [[Existence of Basis of Finite Dimensional Real Vector Space contains Basis (Sifting Lemma)]]. | | | | | [[Replacing a Basis Element with a Nonzero Linear Combination Forms a New Basis of Finite Dimensional Real Vector Space (Exchange Lemma)]]. | | | | | [[Cardinality of Any Two Bases of Finite Dimensional Real Vector Space are Equal (Dimension of FDVS is well-defined)]]. | | | [[Dimension of Finite Dimensional Real Vector Space]]. | | [[Dimension of Subspace is Leq to Dimension of Finite Dimensional Real Vector Space]]. | | | | | [[Subspace of Finite Dimensional Real Vector Space Containing as many Linearly Independent Elements as Dimension]]. | | | | | [[Linearly Independent Subset of Finite Dimensional Real Vector Space Containing as Many Elements Dimension Forms Basis]]. | | | | | [[Spanning Set of Finite Dimensional Real Vector Space Containing as Many Elements Dimension Forms Basis]]. | | | | | | | | [[Direct Sum of Real Vector Spaces]]. | | [[Basis of Direct Sum of Real Vector Spaces]]. | | | | | [[Dimension of Direct Sum of Finite Dimensional Real Vector Spaces]]. | | | | | | | | [[Coordinate Map with respect to Basis for Finite Dimensional Real Vector Space]]. | | | | ### 2.2 Linear Maps | Definitions | Lemmas | Main Theorems | Examples | | ------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------- | -------- | | [[Linear maps]]. | [[Linear Map Fixes Zero]]. | | | | | [[Linear Map is Injective iff Kernel Only Contains Zero]]. | | | | [[Image of Linear Map]]. | [[Image of Linear Map is Subspace of Codomain]] | | | | [[Kernel of Linear Map]]. | [[Kernel of Linear Map is Subspace of Domain]]. | | | | [[Linear Isomorphism]]. | [[Inverse of Linear Isomorphism is Linear Isomorphism]]. | [[Linear Isomorphism Preserves Basis]]. | | | | | [[Linear Isomorphism from Finite Dimensional Vector Space Preserves Dimension]]. | | | | | [[Coordinate Map with respect to Basis for Finite Dimensional Real Vector Space is Linear Isomorphism]] | | | | | [[Every Finite Dimension Real Vector Space is Isomorphic to Some Real n-Space]]. | | | | | | | | [[Left Multiplication Linear Map of Real Matrix]] | [[Left Multiplication Linear Map of Real Matrix is Linear Map]]. | [[Elementary Row Operations Preserve Kernel of Left Multiplication Linear Map of Real Matrix]]. | | | | [[Kernel of Left Multiplication Linear Map of Real Matrix]] | [[Elementary Row Operations Preserve Dimension of Image of Left Multiplication Linear Map of Real Matrix]]. | | | | [[Image of Left Multiplication Linear Map of Real Matrix Equals Column Span of Real Matrix]]. | [[Elementary Column Operations Preserve Dimension of Kernel of Left Multiplication Linear Map]]. | | | | [[Left Multiplication Linear Map of Real Matrix Product]] | [[Elementary Column Operations Preserve Image of Left Multiplication Linear Map of Real Matrix]]. | | | | [[Left Multiplication Linear Map of Real Identity Matrix is Identity Function]]. | [[Smith Normal Form Preserves Dimension of Kernel Left Multiplication Linear Map of Real Matrix]]. | | | | [[Left Multiplication Linear Map of Real Identity Matrix is Identity Function]]. | [[Reduced Row Echelon Form Preserves Solutions to Linear Homogenous System]]. | | | | [[Linear Isomorphism gives Isomorphism Between Subspace of Domain and Image of Subspace]]. | [[Reduced Column Echelon Form Preserves Column Span of Real Matrix]]. | | | | | [[Uniqueness of Smith Normal Form of Real Matrix]]. | | | | | | | | | [[Linear Maps from Finite Dimension Real Vector Space that agree on Basis Elements are Equal]]. | [[Linear Map from Finite Dimension Real Vector Space is Defined by Specifying Images of Basis Elements]]. | | | | | [[Matrix representations of linear map]]. | | | | | [[Matrix representations of linear map]] | | | | | | | | [[Commutative Square]]. | [[Linear Map Between Finite Dimensional Real Vector Spaces gives Commutative Square]]. | | | | | [[Composing Linear Map Between Finite Dimensional Real Vector Spaces gives Commutative Square]] | | | | | | | | | [[Rank of Linear Map]] | | [[Rank-nullity formula]] (Proofs 1 & 2). | | | [[Linear maps]]. | | | | | | | | | | | | [[Dimension of Sum of Finite Dimensional Vector Subspaces (Dimension Formula)]]. | | | [[Existence of complement of subspace of finite-dimensional vector space]]. | | [[Dimension of Finite Dimensional Vector Space in Terms of Dimension of Subspace and Complement]]. | | | | [[Vector Space Elements can Be Written Uniquely as Sum of Subspace and Complement Elements]]. | | | | | [[Existence of Complement to Subspace of Finite Dimensional Vector Space]]. | [[Rank-nullity formula]] (Proof 3). | | | | | | | | | | [[Matrix representations of linear map]]. | | | [[Equivalence of Matrices]]. | [[Equivalence of Real Matrices is Equivalence Relation]]. Moreover, in each equivalence class there is a unique matrix in Smith normal form | | | | | | | | | [[Linear maps form vector space]] | | [[Hom(U,V) is Vector Space]]. | | ### 2.3 Euclidean structures on vector spaces | Definitions | Theorems | Examples | | ------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------ | -------- | | [[Orthonormal Subset of Real n-Space]]. | [[Orthonormal Subset of Real n-Space is Linearly Independent]]. | | | | [[Coordinates with respect to Orthonormal Subset of Real n-Space]]. | | | | [[Gram-Schmidt orthogonalisation in real n-space]]. Note that we can use this to extend any orthonormal set to an orthonormal basis. | | | [[Orthogonal Complement of Subspace of Real n-Space]]. | Orthoganal complements | | | | | | | [[Inner products]]. | | | | [[Euclidean spaces]]. | | | | [[Euclidean Metric on Real n-Space]]. | | | | [[Angle Between Nonzero Euclidean Space Elements]]. | [[Inner Product is At Most Product of Vector Norms (Cauchy-Schwartz Inequality)]]. | | | [[Orthonormal Subset of Euclidean Space]]. | | | | [[Orthogonal Projection in Euclidean Space]]. | [[Gram-Schmidt orthogonalisation in Euclidean space]]. | | | | Orthoganal complements | | | | | | | | [[Coordinate Map with respect to Orthonormal Basis of Euclidean Space Matches Inner Product with Dot Product of Real n-Space]]. | | | | | | # 3. The Structure of Linear maps ### 3.1 Eigenvalues and eigenvectors | Definitions | Lemmas | Main Theorems | Examples | | --------------------------------------------------------------------------------------------------------------------------------- | ------ | ----------------------------------------------------------------------------------------------------------------------------- | -------- | | [[Linear maps]]. | | | | | [[Eigenpair of Linear Operator]]. | | | | | [[Eigenpair of Real Square Matrix]]. | | | | | | | | | | [[Determinant]]. | 1 | [[Determinant of Real Square Matrix is Zero iff Kernel Contains Nonzero Element]]. | | | | 2 | [[Determinant of Product of Real Square Matrices]]. | | | | 3 | [[Existence of Determinant of Real 2 x 2 Matrix]]. | | | | 4 | [[Existence of Determinant of Real 3 x 3 Matrix]]. | | | | 5 | [[Leibniz Formula of Determinant of Real Square Matrix]]. | | | [[Minor of Real Square Matrix]]. | | [[Cofactor Expansion of Determinant of Real Square Matrix]]. | | | [[Cofactor of Real Square Matrix]]. | | [[Determinants of Transpose and Real Square Matrix are Equal]]. | | | | | | | | [[Characteristic Polynomial of Real Square Matrix]]; [[Algebraic Multiplicity of Eigenvalue of Real Square Matrix]] | | | | | [[Eigenspace of Real Square Matrix Corresponding to Eigenvalue]]; [[Geometric Multiplicity of Eigenvalue of Real Square Matrix]]. | | [[Eigenvectors Corresponding to Distinct Eigenvalues of Real Square Matrix are Linearly Independent]]. | | | | | | | | [[Similar Real Square Matrices]]. | | [[Similar Real Square Matrices have The Same Eigenvalues]]. | | | [[Diagonalisable Real Square Matrix]]. | | [[Real Square Matrix of Order n is Diagonalisable iff it has n Linearly Independent Eigenvectors (Diagonalisation Theorem)]]. | | | | | [[Real Square Matrix with n Distinct Eigenvalues is Diagonalisable]]. | | | | | [[Determinant of Diagonalisable Real Square Matrix is Product of Eigenvalues]]. | | ### 3.2 Orthogonal and symmetric matrices | Definitions | Theorems | Examples | | ---------------------------------- | ------------------------------------------------------------------------------------------------------- | -------- | | [[Orthogonal endomorphisms of Euclidean spaces]]. | [[Real Square Matrix is Orthonormal iff its Rows (Columns) Form an Orthonormal Basis]]. | | | | | | | *[[Symmetric Real Square Matrix]]. | [[Eigenvectors Corresponding to Distinct Eigenvalues of Symmetric Real Square Matrix are Orthogonal]]. | | | | [[Symmetric Real Square Matrix with Real Distinct Eigenvalues is Diagonalisable by Orthogonal Matrix]]. | | | | [[Eigenvalues of Symmetric Real Square Matrix are Real]]. | |