Notation
Note that numbered subreferences like “Definition 1.2” refer to Topic 1, Definition 2.
\(S^+\) is the positive subset of \(S\). Set complements are denoted \(A\setminus B\).
We notate a vector in boldface as \(\vb v = \mqty(1\\2\\3) = (1,2,3)^T \in \Z^3\). The generic field \(\F\) is either \(\R\) or \(\C\).
The set of matrices with \(m\) rows and \(n\) columns with elements from \(\F\) is \(M_{m\times n}(\F)\).
\[A = \pmqty{ {(A)}_{11} & {(A)}_{12} & \dotsb & {(A)}_{1n} \\ {(A)}_{21} & {(A)}_{22} & \dotsb & {(A)}_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ {(A)}_{m1} & {(A)}_{m2} & \dotsb & {(A)}_{mn} } = \pmqty{ a_{11} & a_{12} & \dotsb & a_{1n} \\ a_{21} & a_{22} & \dotsb & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dotsb & a_{mn} } = \pmqty{\vb A^1 \\ \vb A^2 \\ \vdots \\ \vb A^m} = (\vb a_1, \vb a_2,\dotsc,\vb a_n)\]A row vector is a \(1\times n\) matrix \(\vb M = (\vb M_1,\vb M_2,\vb M_3) \in M_{1 \times 3}\). A column vector is a \(n\times 1\) matrix, notice that \(M_{n\times 1}(\F) = \F^n\).
Definitions
| Name (Reference) | Statement | |
|---|---|---|
Linearity |
\(F:A \to B\), \(x,y \in A\), scalar \(z\) |
|
Dot Product in \(\R^n\) (2.1) |
\(\vb v = (v_1,\dotsc,v_n)^T\), \(\vb w = (w_1,\dotsc,w_n)^T \in \R^n\) |
\[\vb v\vdot\vb w = v_1 w_1 + \dotsb + v_n w_n\] |
Angle (2.6) |
\[\vb v, \vb w \in \R^n\] | \[\theta = \arccos(\dfrac{\vb v\vdot\vb w}{\norm{\vb v}\norm{\vb w}}) \in [0,\pi]\] |
Standard Inner Product in \(\F^n\) (3.5) |
\(\vb v = (v_1,\dotsc,v_n)^T \in \F^n\), \(\vb w = (w_1,\dotsc,w_n)^T \in \F^n\) |
\[\ip{\vb v}{\vb w} = v_1 \bar w_1 + \dotsb + v_n \bar w_n\] |
Length (2.3/3.2) |
\[\vb v\in\F^n\] | \(\norm{\vb v} = \sqrt{\ip{\vb v}{\vb v}}\) |
Unit Vector (2.4) |
\[\vu v\in\F^n\] | \[\norm{\vu v} = 1\] |
Orthogonality (2.5/3.3) |
\[\vb u, \vb v\in\F^n\] | \[\ip{\vb u}{\vb v} = 0\] |
Projection (2.7/3.4) |
\(\vb v,\vb w\in\F^n\), \(\vb v \neq 0\) |
\[\Proj_{\vb w}(\vb v) = \dfrac{\ip{\vb v}{\vb w}\vb w}{\norm{\vb w}^2} = \ip{\vb v}{\vu w}\vu w\] |
(Scalar) Component (2.8) |
\(\vb v,\vb w\in\F^n\), \(\vb v \neq 0\) |
\[\norm{\vb v}\cos\theta\] |
Remainder (2.9) |
\(\vb z,\vb w\in\F^n\), \(\vb w \neq 0\) |
\[\Perp_{\vb w}(\vb v) = \vb v - \Proj_{\vb w}(\vb v)\] |
Cross Product (4.1) |
\(\vb u = (u_1,u_2,u_3)^T\), \(\vb v = (v_1,v_2,v_3) \in \R^3\) |
\[\vb u \cp \vb v = \mqty(u_2 v_3 - u_3 v_2 \\ -{(u_1 v_3 - u_3 v_1)} \\ u_1 v_2 - u_2 v_1) = \det(\mqty(\vb i&\vb j&\vb k\\u_1&u_2&u_3\\v_1&v_2&v_3))\] |
Linear Combination (5.1) |
\(\vb v_1,\dotsc,\vb v_p \in \F^n\), \(a_1,\dotsc,a_p \in \F\) |
\[a_1 \vb v_1 + \dotsb + a_p \vb v_p\] |
Span (5.2) |
\[\vb v_1,\dotsc,\vb v_p \in \F^n\] | \[\Span(\\{\vb v_1,\dotsc,\vb v_p\\}) = \\{a_1 \vb v_1 + \dotsb + a_p \vb v_p : a_1,\dotsc,a_p \in \F \\}\] |
Line (6A.3) |
\(\vb v,\vb w\in\R^n\), \(\vb w \neq 0\) |
\(L = \\{ \vb v + t\vb w : t\in\R \\}\). If \(\vb v \in \Span(\\{ \vb w \\})\), then \(L = \Span(\\{ \vb w \\})\) |
Vector Equation of a Line (6A.2/6A.4) |
\(\vb v,\vb w\in\R^n\), \(\vb w \neq 0\) |
\[\vb x = \vb v + t\vb w\] |
Parametric (Scalar) Equations of a Line (6A.1/6A.5) |
\(\vb v = (v_1,\dotsc,v_n)^T\), \(\vb w = (w_1,\dotsc,w_n)^T\in\R^n\), \(\vb w \neq 0\) |
\[\begin{cases}x_1=v_1+t w_1 \\ \quad\quad\vdots \\ x_n=v_n+t w_n\end{cases}\] |
Plane (6B.7/6B.9) |
\(\vb p, \vb v, \vb w \in \R^n\), \(\vb v \neq 0\), \(\vb w \not\in \Span(\\{ \vb v \\})\) |
\(\Pi = \\{ \vb p + s\vb v + t\vb w : s,t\in\R \\}\). If \(\vb p = \vb 0\), then \(\Pi = \Span(\\{ \vb v, \vb w \\})\) |
Vector Equation of a Plane (6B.8/6B.10) |
\(\vb p, \vb v, \vb w \in \R^n\), \(\vb v \neq 0\), \(\vb w \not\in \Span(\\{ \vb v \\})\) |
\[\vb x = \vb p + s\vb v + t\vb w\] |
Scalar Equation of a Plane in \(\R^3\) (6B.11) |
\(\vb p, \vb v, \vb w\in\R^3\), \(\vb v \neq 0\), \(\vb w \not\in \Span(\\{ \vb v \\})\) |
\[(\vb v \cp \vb w) \vdot (\vb x - \vb p) = 0\] |
Linear System (7A.2/7A.3) |
\(a_{11},\dotsc,a_{mn}\in\F\) (coefficients), \(b_1,\dotsc,b_m\in\F\) (RHS), \(x_1,\dotsc,x_n\in\F\) (unknowns) |
\[(*)\;\left\{\begin{align} a_{11}x_1 + a_{12}x_2 + \dotsb + a_{1n}x_n & = b_1 & (e_1) \\ a_{21}x_1 + a_{22}x_2 + \dotsb + a_{2n}x_n & = b_2 & (e_2) \\ \vdots\quad\quad\quad\quad \\ a_{m1}x_1 + a_{m2}x_2 + \dotsb + a_{mn}x_n & = b_m & (e_m) \end{align}\right.\] |
Solution to a Linear System (7A.4/7A.5) |
Linear system |
|
Consistency (7A.6) |
Linear system |
The solution set is non-empty (otherwise inconsistent) |
Equivalent (7A.7) |
Two linear systems |
The solution sets are equal |
Elementary Operations (7B.8) |
Distinct equations \(e_i\) and \(e_j\) |
|
Triviality (7.B9) |
Equation \(e_i\) |
\(e_i \equiv 0 = 0\) (otherwise non-trivial) |
Coefficient Matrix (8A.1) |
Linear system |
\[A = \mqty( a_{11}&a_{12}&\dotsb&a_{1n} \\ a_{21}&a_{22}&\dotsb&a_{2n} \\ \vdots&\vdots&\ddots&\vdots \\ a_{m1}&a_{m2}&\dotsb&a_{mn} ) \in M_{m\times n}\] |
Augmented Matrix (8A.2) |
Linear system |
\[B = (A\vert\vb{b}) = \amat{cccc\\|c}{ a_{11} & a_{12} & \dotsb & a_{1n} & b_1 \\ a_{21} & a_{22} & \dotsb & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dotsb & a_{mn} & b_n } \in M_{m\times(n+1)}\] |
Row Echelon Form (8B.7) |
\[A\in M_{m\times n}\] |
|
Row-Reduced Echelon Form (8B.11) |
\[\REF(A)\] |
|
Rank (9.1) |
\[A\in M_{m\times n}\] | \(\rank(A)\) is the number of pivots in \(\RREF(A)\) |
Nullity (9.2) |
\[A\in M_{m\times n}\] | \[\nullity(A) = n - \rank(A)\] |
Homogenous (10A.1) |
System \((A\vert\vb b)\) |
\[\vb b = \vb 0\] |
Nullspace (10A.2) |
Coefficient matrix \(A\) |
\[N(A) = \\{ \vb x : A\vb x = \vb 0 \\}\] |
Matrix-Vector Multiplication (11A.2a) |
\(A\in M_{m\times n}\), \(\vb x \in \F^n\) |
\(A\vb x = \mqty( a_{11}x_1 + a_{12}x_2 + \dotsb + a_{1n}x_n \\ a_{21}x_1 + a_{12}x_2 + \dotsb + a_{2n}x_n \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \dotsb + a_{mn}x_n )\) or \({(A\vb x)}_i = \sum_{j=1}^n a_{ij} x_j\) |
Matrix-Vector Multiplication by Columns (11A.2b) |
\(A\in M_{m\times n}\), \(\vb x \in \F^n\) |
\[A\vb x = x_1\vb a_1 + x_2\vb a_2 + \dotsb + x_n \vb a_n\] |
Matrix-Vector Multiplication by Rows (11A.2c) |
\(A\in M_{m\times n}\), \(\vb x \in \F^n\) |
\[A\vb x = \mqty( \vb A^1_1 x_1 + \vb A^1_2 x_2 + \dotsb + \vb A^1_n x_n \\ \vb A^2_1 x_1 + \vb A^2_2 x_2 + \dotsb + \vb A^2_n x_n \\ \vdots \\ \vb A^m_1 x_1 + \vb A^m_2 x_2 + \dotsb + \vb A^m_n x_n )\] |
Matrix Multiplication (11C.7) |
\(A\in M_{m\times n}\), \(B\in M_{n\times p}\) |
\[AB = (A\vb b_1, A\vb b_2, \dotsc, A\vb b_p)\] |
Column Space (11C.8) |
\[A \in M_{m \times n}\] | \[\Col(A) = \Span(\{ \vb a_1, \vb a_2, \dotsc, \vb a_n \})\] |
Square Matrix (12B.5) |
\[A \in M_{m\times n}\] | \[m = n\] |
Symmetric (12B.6) |
\[A \in M_{n\times n}\] | \[A = A^T\] |
Skew-Symmetric (12B.6) |
\[A \in M_{n\times n}\] | \[A = -A^T\] |
Upper Triangular (\(U\sym\), 12B.7) |
\[A \in M_{n\times n}\] | \(a_{ij} = 0\) if \(i > j\) |
Lower Triangular (\(L\sym\), 12B.8) |
\[A \in M_{n\times n}\] | \(a_{ij} = 0\) if \(i < j\) |
Diagonal (12B.9) |
\[A \in M_{n\times n}\] | \(a_{ij} = 0\) if \(i \neq j\) |
Main Diagonal (12B.10) |
\[A \in M_{n\times n}\] | \[(a_{11}, a_{22}, \dotsc, a_{nn})\] |
Identity Matrix (12B.11) |
\(I_n = \mqty(1&0&\cdots&0\\0&1&\cdots&0\\\vdots&\vdots&\ddots&0\\0&0&0&1)\) and \(AI_n = I_nA = A\) for all \(A\in M_{n\times n}\) |
|
Transformation of a Matrix (13A.1) |
\[A \in M_{n\times n}\] | \(T_A : \F^n \to \F^m\), \(T_A(\vb x) = A\vb x\) |
Linear Transformation (13A.2) |
\[F : \F^n \to \F^m\] | \(F\) is linear |
Range (13A.3) |
\[T : \F^n \to \F^m\] | \[R(T) = \{ T(\vb x) : \vb x \in \F^n \}\] |
Nullspace (13A.5) |
\[T : \F^n \to \F^m\] | \[N(T) = \{ \vb x \in \F^n : T(\vb x) = \vb 0_{\F^m} \}\] |
Onto (13A.4) |
\[T : \F^n \to \F^m\] | \[R(T) = \F^m\] |
One-to-One (13A.6) |
\[T : \F^n \to \F^m\] | \[\vb x \neq \vb y \implies T(\vb x) \neq T(\vb y)\] |
Matrix of a Transformation (13B.7) |
\(T : \F^n \to \F^m\), basis \(S\) |
\[[T]_S = \pqty{T(\vb e_1), T(\vb e_2), \dotsc, T(\vb e_n)}\] |
Invertibility of Matrix (13C.10) |
\(A \in M_{n \times n}\), \(B \in M_{n \times n}\), \(AB=BA=I_n\) |
\[A^{-1} = B\] |
Singular Matrix (13C.11) |
\[A \in M_{n \times n}\] | \(A\) is not invertible |
Submatrix (15A.1) |
\[A \in M_{n \times n}\] | \(M_{ij}(A) \in M_{(n-1)\times(n-1)}\) missing \(i\)th row and \(j\)th column |
Determinant (15A.2) |
\(A \in M_{1 \times 1}\), \(B \in M_{2\times 2}\) |
\(\det(A) = a_{11}\), \(\det(B) = a_{11}a_{22} - a_{12}a_{21}\) |
Cofactor (15A.6) |
\[A \in M_{n \times n}\] | \[C_{ij}(A) = (-1)^{i+j} \det(M_{ij}(A))\] |
\(I\)th Row Expansion (15A.4) |
\(A \in M_{n \times n}\), \(I \leq n\) |
\[\det(A) = \sum_{j=1}^n a_{Ij} C_{Ij}(A)\] |
\(J\)th Column Expansion (15A.5) |
\(A \in M_{n \times n}\), \(J \leq n\) |
\[\det(A) = \sum_{i=1}^n a_{iJ} C_{iJ}(A)\] |
Adjoint/Adjunct Matrix (15C.7) |
\[A \in M_{n \times n}\] | \((\adj(A))_{ij} = C_{ji}(A)\) (transpose of cofactor matrix) |
Eigenvector (16A.1) |
\(A \in \Mnn\), \(\vb x \neq \vb 0\), \(\lambda \in \F\) |
\[A\vb x = \lambda \vb x\] |
Eigenvalue Equation (16A.2) |
\(A \in \Mnn\), \(\vb x \neq \vb 0\), \(\lambda \in \F\) |
\[(A - \lambda I)\vb x = \vb 0\] |
Characteristic Polynomial (16A.3) |
\[A \in \Mnn\] | \(\Delta_A(t) = \det(A - tI) \in \F[t]\) (roots are eigenvalues) |
Eigenspace (16A.4) |
\(A \in \Mnn\), \(\lambda_1\) eigenvalue of \(A\) |
\[E_{\lambda_1} = N(A - \lambda_1 I)\] |
Similarity (16B.5) |
\[A,B \in \Mnn\] | \(\exists Q \in \Mnn\), \(Q^{-1}AQ = B\) |
Similarity Transformation (16B.6) |
\[Q \in \Mnn\] | \(T: \Mnn \to \Mnn\), \(T(A) = Q^{-1}AQ\) |
Trace (16B.7) |
\[A \in \Mnn\] | \[\tr(A) = \sum_{i=1}^n a_{ii}\] |
Diagonalizable (16B.8) |
\[A \in \Mnn\] | \(\exists D\) diagonal matrix, \(A\) similar to \(D\) |
Subspace (17A.1) |
\[V \subseteq \F^n\] |
|
Trivial Subspace (Example 17A.1) |
\(V = \{\vb 0\}\) and \(V = \F^n\) |
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Linear (In)dependence (17A.2/17A.3) |
\[\vb v_1,\vb v_2,\dotsc,\vb v_p\] | \(\exists c_1,c_2,\dotsc,c_p \in \F\setminus\{0\}\), \(c_1\vb v_1 + \dotsb + c_p\vb v_p = \vb 0\) |
Basis (17A.4) |
\(B = \{\vb v_1,\vb v_2,\dotsc,\vb v_p\}\) basis for \(V\) |
|
Dimension (17C.5) |
\(B\) basis for \(V\) |
\[\dim(V) = \abs{B}\] |
Standard Basis (17C.6) |
\(S = \{ \vb e_1,\dotsc,\vb e_n \} \subset \F^n\), \((\vb e_1,\dotsc,\vb e_n) = I_n\) |
|
Coordinates and Components (17D.7) |
\(B = \{ \vb v_1,\dotsc,\vb v_n \}\) basis for \(\vb v \in \F^n\) |
\([\vb v]_B = (c_1,\dotsc,c_n)^T\), \(\sum c_i \vb v_i = \vb v\) |
Change-of-Basis Matrix (17D.8) |
\(B_1 = \{ \vb v_1,\dotsc,\vb v_n \}\), \(B_2\) bases |
\[{}_{B_2}[I]_{B_1} = ([\vb v_1]_{B_2},\dotsc,[\vb v_n]_{B_2})\] |
Linear Operator (18.1) |
\(T : \F^n \to \F^m\) linear |
\(n = m\), so \([T]\) is square |
Matrix Representation (18.2) |
\(T : \F^n \to \F^n\), basis \(B = \{\vb v_1,\dotsc,\vb v_n\}\) |
\[[T]_B = (T(\vb v_1),\dotsc,T(\vb v_n))\] |
Algebraic Multiplicity (19B.4) |
\(A \in \Mnn\), \(\lambda\) eigenvalue |
\[a_\lambda = \max\{a \in \N : (t - \lambda)^a \mid \Delta_A(t)\}\] |
Geometric Multiplicity (19B.5) |
\(A \in \Mnn\), \(\lambda\) eigenvalue |
\[g_\lambda = \dim(E_\lambda)\] |
Properties of \(\{\vb 0\}\) (20.1) |
\(\Span(\varnothing) = \{\vb 0\}\) so \(\varnothing\) is a basis for \(\{\vb 0\}\) and \(\dim(\{\vb 0\}) = 0\) |
|
Vector Space (21A.1) |
|
For all \(\vb v,\vb w,\vb z \in V\) and \(c,d\in\F\), there is closure:
and eight axioms hold:
|
Rowspace (22.1) |
\[A \in M_{m\times n}\] | \[\operatorname{Row}(A) = \Span(\{\vb A^1,\vb A^2,\dotsc,\vb A^m\}) \subset M_{1\times n}\] |
Matrix Representation of Linear Transformation (23.2) |
\(T : U \to V\), basis \(B_1 = \{\vb u_i\}\) of \(U\) and \(B_2\) of \(V\) |
\[{}_{B_2}[T]_{B_1} = ([T(\vb u_i)]_{B_2}))\] |
Theorems
| Name (Reference) | Statement | ||
|---|---|---|---|
Properties of Complex Conjugation (Lemma I3.4) |
\[z,w\in\C\] | \( \implies \) |
|
Properties of the Modulus (Lemma I3.5) |
\[z,w\in\C\] | \( \implies \) |
|
Properties of Zero (Lemma 1.4) |
\(\vb v \in \F^n\), \(a \in \F\), \(a\vb v = \vb 0\) |
\( \implies \) | \(\vb v = \vb 0\) or \(a = 0\) |
Properties of the Dot Product (Lemma 2.1) |
\(\vb v,\vb w,\vb z\in\R^n\) and \(a\in\R\) |
\( \implies \) |
|
Properties of the Standard Inner Product on \(\C^n\) (Lemma 3.1) |
\(\vb v,\vb w,\vb z\in\C^n\) and \(a\in\C\) |
\( \implies \) |
|
Properties of the Length (Lemma 2.2/3.2) |
\(\vb v\in\F^n\) and \(a\in\F\) |
\( \implies \) |
|
Properties of the Cross Product (Lemma 4.1) |
\(\vb u,\vb v\in\R^3\) and \(\vb z = \vb u \cp \vb v\) |
\( \implies \) |
|
Solution Set to a Linear System (Theorem 7A.1) |
Solution set \(S\) to system \((*)\) |
\( \implies \) | \[\abs{S} \in \\{0, 1, \abs{\F}\\}\] |
Lemma 9.1 |
System \((A\vert\vb{b})\) is consistent |
\( \iff \) | \[\rank(A) = \rank(A\vert\vb{b})\] |
Rank-Nullity Theorem (Lemma 9.2) |
\(A\in M_{m\times n}\) is consistent |
\( \implies \) | Solution set has \(\nullity(A)\) parameters. |
Linearity of Matrix Multiplication (Lemma 11A.1) |
\(A \in M_{m\times n}\), \(\vb x,\vb y\in\F^n\), \(c\in\F\) |
\( \implies \) |
|
Lemma 11B.2 |
Homogenous solution \(S\), \(\vb x_1, \vb x_2 \in S\), \(c\in\F\) |
\( \implies \) |
|
Relation between \(\tilde S\) and \(S\) (Lemma 11B.3/11B.4) |
Homogenous solution \(S\), inhomogeneous solution \(\tilde S\) |
\( \implies \) |
|
Relation between inhomogeneous solutions (Lemma 11B.5) |
Inhomogeneous solutions \(\vb p_1 \in \tilde S_1\) and \(\vb p_2 \in \tilde S_2\) |
\( \implies \) |
|
Lemma 11C.6 |
System \(A\vb x = \vb b\) is consistent |
\( \iff \) | \[\vb b \in \Col(A)\] |
Properties of Matrix Addition (Lemma 12A.1) |
\[A,B,C \in M_{m \times n}\] | \( \implies \) |
|
Properties of Matrix-Scalar Multiplication (Lemma 12A.2) |
\(A,B\in M_{m\times n}\), \(C\in M_{n\times p}\), \(c,d\in\F\) |
\( \implies \) |
|
Properties of the Transpose (Lemma 12A.3) |
\(A,B\in M_{m\times n}\), \(c\in\F\) |
\( \implies \) |
|
Properties of Matrix Multiplication (Lemma 12A.4) |
\(A,G\in M_{m\times n}\), \(B,D\in M_{n\times p}\), \(C\in M_{p\times q}\) |
\( \implies \) |
|
Zero Under Linearity (Lemma 13A.3) |
\(T: \F^n \to \F^m\) linear |
\( \implies \) | \[T(\vb 0_{\F^n}) = \vb 0_{\F^m}\] |
Range of Matrix Function (Lemma 13A.4) |
\[A \in M_{m\times n}\] | \( \implies \) | \[R(T_A) = \Col(A)\] |
Onto Matrix Characterization (Corollary 13A.1/13A.2) |
\(A \in M_{m\times n}\) is onto |
\( \iff \) | \[\Col(A) = \F^m \iff \rank(A) = m\] |
One-to-One Matrix Characterization (Lemma 13A.6/Corollary 13A.3) |
\(A \in M_{m\times n}\) is one-to-one |
\( \iff \) | \(N(T_A) = \{ \vb 0_{\F^n} \}\) \(\iff\) \(\nullity(A) = 0\) \(\iff\) \(\rank(A) = n\) |
Matrix/Transformation Determination (Remarks 13B.3/13B.4, Lemma 13B.7) |
\(A \in M_{m\times n}\), \(T : \F^n \to \F^m\) |
\( \implies \) |
|
Linearity of Composite Function (Lemma 13C.9) |
\(T_1 : \F^n \to \F^m\), \(T_2 : \F^m \to \F^p\) |
\( \implies \) | \(T_2 \circ T_1\) is linear |
Matrix of Composite Function (Lemma 13C.10) |
\(T_1 : \F^n \to \F^m\), \(T_2 : \F^m \to \F^p\) |
\( \implies \) | \[[T_2 \circ T_1]_S = [T_2]_S [T_1]_S\] |
Unique Inverse (Lemma 13C.11) |
\(A,B \in M_{n \times n}(\F)\), \(AB=BA=I_n\) |
\( \implies \) | \(B\) is unique |
Lemma 13C.12 |
\(A \in M_{n \times n}\) invertible |
\( \implies \) | \(A\vb x = \vb b\) has solution \(\vb x = A^{-1}\vb b\) |
Properties of Inverse (Lemma 13C.13) |
\(A,B \in M_{n \times n}\) and invertible |
\( \implies \) |
|
Left/Right Inverse (Lemma 14.1) |
\(A,B \in M_{n \times n}\), \(AB = I_n\) |
\( \iff \) | \[BA = I_n\] |
Invertibility of Matrix (Lemma 14.2) |
\(A \in M_{n \times n}\) invertible |
\( \iff \) | \[\rank(A) = n\] |
Determinant of Transpose (Lemma 15A.1) |
\[A \in M_{n \times n}\] | \( \implies \) | \[\det(A^T) = \det(A)\] |
Determinant of Triangular Matrix (Lemma 15A.2) |
\(A \in M_{n \times n}\), \(U\sym\) or \(L\sym\) |
\( \implies \) | \[\det(A) = \prod a_{ii}\] |
Properties of Determinant (Theorem 15A.1) |
\[A \in M_{n \times n}\] | \( \implies \) |
|
EROs and the Determinant (Corollary 15B.4) |
\(A \in M_{n \times n}\) and \(B\) is \(A\) after an ERO of type |
\( \implies \) |
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Corollary 15B.5 |
\(A \in M_{n \times n}\) and \(B\) is \(A\) after elementary matrices \(E_1,\dotsc,E_q\) |
\( \implies \) | \[\det(B) = \det(E_q \cdots E_1 A) = \det(E_q)\cdots\det(E_1)\det(A)\] |
Determinants and Invertibiliy (Corollary 15B.7) |
\(A \in M_{n \times n}\) invertible |
\( \iff \) | \[\det(A) \neq 0\] |
Determinant of Product (Corollary 15B.8) |
\[A,B \in M_{n \times n}\] | \( \implies \) | \[\det(AB) = \det(A)\det(B)\] |
Determinant of Inverse (Corollary 15B.9) |
\(A \in M_{n \times n}\) invertible |
\( \implies \) | \[\det(A^{-1}) = \det(A)^{-1}\] |
Lemma 15C.3 |
\[A \in M_{n \times n}\] | \( \implies \) | \[A\adj(A) = \adj(A)A = \det(A)I_n\] |
Corollary 15C.10 |
\(A \in M_{n \times n}\) invertible |
\( \implies \) | \[A^{-1} = \frac{1}{\det(A)}\adj(A)\] |
Cramer’s Rule (Lemma 15C.4) |
\(A \in M_{n \times n}\) invertible, \(A\vb x = \vb b\) |
\( \implies \) | \(x_j = \frac{\det(B_j)}{\det(A)}\) where \(B_j\) is \(A\) with \(\vb a_j = \vb b\) |
Parallelogram Area (Lemma 15C.5) |
Parallelogram bounded by \(\vb v = (v_1,v_2)^T\), \(\vb w = (w_1,w_2)^T\) |
\( \implies \) | \[A = \abs{\det(\mqty(v_1&v_2\\w_1&w_2))}\] |
Scalar Triple Product (Lemma 15C.6) |
Parallelipiped bounded by \(\vb x\), \(\vb y\), \(\vb z\) |
\( \implies \) | \[V = STP(\vb x,\vb y,\vb z) = \vb x \vdot (\vb y \cp \vb z) = \det((\vb x,\vb y,\vb z))\] |
Similarity Invariants (Lemma 16B.1) |
\(A\) and \(B\) similar |
\( \implies \) | \(\det(A) = \det(B)\) and \(\tr(A) = \tr(B)\) |
Diagonalization I (Lemma 16B.2) |
\(A \in \Mnn\) with distinct eigenpairs \((\lambda_1,\vb v_1),\dotsc,(\lambda_n,\vb v_n)\) |
\( \implies \) | If \(P = (\vb v_1,\dotsc,\vb v_n)\), \(P^{-1}AP = \diag(\lambda_1,\dotsc,\lambda_n)\) |
Properties of the Characteristic Polynomial (Lemma 16B.3) |
\[A \in \Mnn\] | \( \implies \) |
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Properties of the Complex \(\Delta_A\) (Lemma 16B.4) |
\[A \in \Mnn(\C)\] | \( \implies \) |
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Lemma 16B.5 (\(\R\)) / Corollary 16B.1 (\(\C\)) |
\(A \in \Mnn\) invertible |
\( \iff \) | \(0\) not an eigenvalue of \(A\) |
Lemma 16B.6 |
\(A,B \in \Mnn\) similar |
\( \implies \) | \[\Delta_A(t) = \Delta_B(t)\] |
Subspace Characterization (Lemma 17A.1) |
\(V \subseteq \F^n\) is a subspace |
\( \iff \) |
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Example 17A.1 |
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Lemma 17B.2 |
\[0 \in S \subseteq \F^n\] | \( \implies \) | \(S\) linearly dependent |
Lemma 17B.3 |
\(S = \{\vb x\}\) is linearly dependent |
\( \iff \) | \[\vb x = \vb 0\] |
Lemma 17B.4 |
\(\abs{S} = 2\) is linearly dependent |
\( \iff \) | One vector is a scalar multiple of the other |
Lemma 17B.5 |
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\( \implies \) |
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Corollary 17B.1 |
\(S \subseteq \F^n\), \(\abs{S} > n\) |
\( \implies \) | \(S\) linearly dependent |
Lemma 17B.6 |
\(S\) linearly independent, \(S \cup \{\vb w\}\) linearly dependent |
\( \iff \) | \[\vb w \in \Span(S)\] |
Lemma 17B.7 |
\(S\) linearly independent, \(\vb v \in S\) |
\( \implies \) | \(S \setminus \{\vb v\}\) linearly independent |
Lemma 17C.8 |
\(V\) subspace of \(\F^n\), \(S \subset V\) |
\( \implies \) | \(\Span(S)\) subspace of \(V\) |
Lemma 17C.9 |
\[\Span(\{\vb v_1,\dotsc,\vb v_p\}) = \F^n\] | \( \iff \) | \[\rank((\vb v_1,\dotsc,\vb v_p)) = n\] |
Lemma 17C.10 |
Basis \(S\) for \(\F^n\) |
\( \implies \) | \[\abs{S} = n\] |
Lemma 17C.11 |
\(S \subset \F^n\) linearly independent with \(\abs{S} = n\) |
\( \iff \) | \[\Span(S) = \F^n\] |
Unique Representation (Theorem 17C.1) |
\(B = \{\vb v_i\}\) basis for \(\F^n\), \(\vb x \in \F^n\) |
\( \implies \) | Unique \(c_i\) where \(\sum c_i\vb v_i = \vb x\) |
Linearity of Coordinates (Lemma 17D.12) |
\(B\) basis for \(\F^n\) |
\( \implies \) | \([\ ]_B : \F^n \to \F^n\), \(\vb x \mapsto [\vb x]_B\) is linear |
Change of Basis (Lemma 17D.13) |
\(B_1 = \{ \vb v_i \}\), \(B_2 = \{ \vb w_i \}\) bases for \(\vb x \in \F^n\) |
\( \implies \) | \([\vb x]_{B_1} = {}_{B_1}[I]_{B_2} [\vb x]_{B_2}\) and vice versa |
Corollary 17D.3 |
\(B_1\), \(B_2\) bases |
\( \implies \) | \[({}_{B_2}[I]_{B_1})^{-1} = {}_{B_1}[I]_{B_2}\] |
Lemma 18.1 |
\(T\) linear operator, \(B\) basis |
\( \implies \) | \[T(\vb v) = [T]_B [\vb v]_B\] |
Lemma 18.2 |
\(T\) linear operator, \(B_1\), \(B_2\) bases |
\( \implies \) | \[[T]_{B_2} = {}_{B_2}[I]_{B_1}[T]_{B_1}{}_{B_1}[I]_{B_2}\] |
Lemma 19A.1 |
\(T\) linear operator, \(B\) basis, \((\lambda,\vb x)\) eigenpair |
\( \iff \) | \((\lambda,[\vb x]_B)\) eigenpair of \([T]_B\) |
Lemma 19A.2 |
\(T\) diagonalizable linear operator |
\( \iff \) | \(\exists B\) basis made of eigenvectors |
Lemma 19A.3 |
\(T\) diagonalizable linear operator, \(B\) basis |
\( \iff \) | \([T]_B\) diagonalizable |
Corollary 19A.1 |
\(A \in \Mnn\) diagonalizable |
\( \iff \) | Eigenvectors of \(A\) are basis |
Lemma 19A.4 |
\(A \in \Mnn\) with distinct eigenvalues \(\lambda_i\) |
\( \implies \) | \(\{\vb v_i\}\) linearly independent |
Lemma 19B.5 |
\(\lambda\) eigenvalue of \(A \in \Mnn\) |
\( \implies \) | \[1 \leq g_\lambda \leq a_\lambda\] |
Lemma 19B.6 |
\(\lambda_i\) eigenvalues of \(A \in \Mnn\) with bases \(B_i\) of eigenspaces \(E_{\lambda_i}\) |
\( \implies \) | \(B = \bigcup B_i\) linearly independent |
Lemma 19B.7 |
\(A \in \Mnn\) where \(\Delta_A(t) = h(t)\prod (\lambda_i-t)^{a_{\lambda_i}}\) with \(h(t)\) irreducible and \(A\) is diagonalizable |
\( \iff \) | \(h(t) = 1\) and \(a_{\lambda_i} = g_{\lambda_i}\) for all \(i\) |
Lemma 20.1 |
\(V\) subspace of \(\F^n\) |
\( \implies \) | \(\exists W \subseteq V\) linearly independent, \(\Span(W) = V\), and \(\abs{W} \leq n\) |
Lemma 20.2 |
\(V\) subspace with \(U\) and \(W\) bases |
\( \implies \) | \[\abs{U} = \abs{W}\] |
Replacement Theorem (Lemma 20.3) |
\(V \neq \{\vb 0\}\) subspace of \(\F^n\) with basis \(W\) |
\( \implies \) | \(\exists B\) basis for \(\F^n\), replacing some standard basis with vectors from \(W\) |
Remark 20.1 |
\[A \in \Mnn\] | \( \implies \) | \[\rank(A) = \dim(\Col(A))\] |
Rank-Nullity Theorem (Theorem 20.1) |
\[A \in \Mnn\] | \( \implies \) | \[\begin{aligned}n &= \dim(\Col(A)) + \dim(N(A))\\ &= \rank(A) + \nullity(A)\end{aligned}\] |
Lemma 21B.1 |
Vector space \((V,\oplus,\F,\odot)\) |
\( \implies \) | \(\vb 0 \in V\) unique |
Lemma 21B.2 |
\[\vb v \in V\] | \( \implies \) | \((-\vb v) \in V\) unique |
Lemma 21B.3 |
\(\vb v \in V\), \(a \in \F\) |
\( \implies \) | \(0\odot\vb v = \vb 0\) and \(a\odot\vb 0 = \vb 0\) |
Lemma 21B.4 |
\[\vb v \in V\] | \( \implies \) | \[(-\vb v) = (-1)\odot\vb v\] |
Lemma 21B.5 |
\(\vb v \in V\), \(a \in \F\), \(a\odot\vb v = \vb 0\) |
\( \implies \) | \(\vb v = \vb 0\) or \(a = 0\) |
Lemma 22.1 |
\(A \in M_{m\times n}\), \(B\) is \(A\) after EROs |
\( \implies \) | \[\operatorname{Row}(A) = \operatorname{Row}(B)\] |
Corollary 22.1 |
\[A \in M_{m\times n}\] | \( \implies \) | \[\dim(\operatorname{Row}(A)) = \rank(A)\] |
Lemma 22.2 |
\[A \in M_{m\times n}\] | \( \implies \) | \[\rank(A) = \rank(A^T)\] |
Lemma 23.1 |
\(T : U \to V\), basis \(B_1\) of \(U\) and \(B_2\) of \(V\) |
\( \) | \[T(\vb x) = {}_{B_2}[T]_{B_1}\,[\vb x]_{B_1}\] |