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 \(z\)

  1. \[F(x+y) = F(x) + F(y)\]
  2. \[F(zx) = zF(x)\]

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)

\(\vb u = (u_1,u_2,u_3)^T\), \(\vb v = (v_1,v_2,v_3) \in \R^3\)

\[\vb u \cp \vb v = \mqty(u_2 v_3 - u_3 v_2 \\ -{(u_1 v_3 - u_3 v_1)} \\ u_1 v_2 - u_2 v_1)\]

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

  1. Single solution \(\vb x = (\vb x_1,\vb x_2,\dotsc,\vb x_n) \in \F^n\)
  2. Solution set \(S \subseteq \F^n\)

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 \(e_j\)

  1. Interchange: \(e_i \harr e_j\)
  2. Multiply: \(e_i \to k e_i\), \(k \in \F \setminus\{0\}\)
  3. Add: \(e_i = e_i + ce_j\), \(c \in \F \setminus \{0\}\)

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)

\[A\in M_{m\times n}\]
  • \(\REF(A)\) is equivalent to \(A\)
  • Zero rows are at the bottom
  • First entry in non-zero row is right of first entry in lower rows

Row-Reduced Echelon Form (8B.11)

\[\REF(A)\]
  • \(\RREF(A)\) is equivalent to \(\REF(A)\)
  • All pivots are 1
  • A pivot is the only value in its column

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)

\(A \in M_{n \times n}\), \(J \leq n\)

\[\det(A) = \sum_{i=1}^n a_{iJ} C_{iJ}(A)\]

Theorems

Name (Reference) Statement

Properties of Complex Conjugation (Lemma I3.4)

\[z,w\in\C\] \( \implies \)
  1. \[\overline{(\bar z)} = z\]
  2. \[\overline{(wz)} = \bar w \bar z\]
  3. \[\overline{(w+z)} = \bar w + \bar z\]
  4. \[z + \bar z = 2\Re(z)\]
  5. \[z - \bar z = 2i\Im(z)\]

Properties of the Modulus (Lemma I3.5)

\[z,w\in\C\] \( \implies \)
  1. \(\abs{z} \geq 0\) and \(\abs{z} = 0 \iff z = 0\)
  2. \[\abs{\bar z} = \abs{z}\]
  3. \[\abs{zw} = \abs{z}\abs{w}\]
  4. \(\abs{z+w} \leq \abs{z} + \abs{w}\) (Triangle Inequality)

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 \(a\in\R\)

\( \implies \)
  1. \(\vb v \vdot \vb w = \vb w \vdot \vb v\) (symmetry)
  2. \[(\vb v + \vb w)\vdot\vb z = \vb v \vdot \vb z + \vb w \vdot \vb z\]
  3. \[(a\vb w)\vdot \vb v = a(\vb w \vdot \vb v)\]
  4. \(\vb v\vdot\vb v \geq 0\) and \(\vb v \vdot \vb v = 0 \iff \vb v = \vb 0\) (non-negativity)

Properties of the Standard Inner Product on \(\C^n\) (Lemma 3.1)

\(\vb v,\vb w,\vb z\in\C^n\) and \(a\in\C\)

\( \implies \)
  1. \(\ip{\vb v}{\vb w} = \overline{\ip{\vb w}{\vb v}}\) (conjugate symmetry)
  2. \[\ip{\vb v + \vb w}{\vb z} = \ip{\vb v}{\vb z} + \ip{\vb w}{\vb z}\]
  3. \[\ip{a\vb w}{\vb v} = a\ip{\vb w}{\vb v}\]
  4. \(\ip{\vb v}{\vb v} \geq 0\) and \(\ip{\vb v}{\vb v} = 0 \iff \vb v = \vb 0\) (non-negativity)

Properties of the Length (Lemma 2.2/3.2)

\(\vb v\in\F^n\) and \(a\in\F\)

\( \implies \)
  1. \[\norm{a\vb v} = \abs{a}\norm{\vb v}\]
  2. \(\norm{\vb v} \geq 0\) and \(\norm{\vb v} = 0 \iff \vb v = \vb 0\)

Properties of the Cross Product (Lemma 4.1)

\(\vb u,\vb v\in\R^3\) and \(\vb z = \vb u \cp \vb v\)

\( \implies \)
  1. \(\vb z\vdot\vb u = 0\) and \(\vb z\vdot\vb v = 0\)
  2. \(\vb v\cp\vb u = -\vb z\) (skew symmetry)
  3. \[\norm{\vb z} = \norm{\vb u}\norm{\vb v}\sin\theta\]
  4. The right-hand rule
  5. Linearity in both arguments (Lemma 4.2)

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)

\(A \in M_{m\times n}\), \(\vb x,\vb y\in\F^n\), \(c\in\F\)

\( \implies \)
  1. \[A(\vb x + \vb y) = A\vb x + A\vb y\]
  2. \[A(c\vb x) = cA\vb x\]

Lemma 11B.2

Homogenous solution \(S\), \(\vb x_1, \vb x_2 \in S\), \(c\in\F\)

\( \implies \)
  1. \[(\vb x_1 + \vb x_2) \in S\]
  2. \[c\vb x_1 \in S\]

Relation between \(\tilde S\) and \(S\) (Lemma 11B.3/11B.4)

Homogenous solution \(S\), inhomogeneous solution \(\tilde S\)

\( \implies \)
  1. \((\vb y_1 - \vb y_2) \in S\) for any \(\vb y_1,\vb y_2 \in \tilde S\)
  2. \(\tilde S = \{ \vb y_p + \vb x : \vb x \in S \}\) given \(\vb y_p \in S\)

Relation between inhomogeneous solutions (Lemma 11B.5)

Inhomogeneous solutions \(\vb p_1 \in \tilde S_1\) and \(\vb p_2 \in \tilde S_2\)

\( \implies \)
  • \[\tilde S_2 = \{ \vb p_2 + (\vb z - \vb p_1) : \vb z \in \tilde S_1 \}\]
  • If \(\tilde S_1 = \{ \vb p_1 + W\vb a : \vb a \in \F^n \}\) then \(\tilde S_2 = \{ \vb p_2 + W\vb a : \vb a \in \F^n \}\)

Lemma 11C.6

System \(A\vb x = \vb b\) is consistent

\( \iff \) \[\vb b \in \Col(A)\]

Properties of Matrix Addition (Lemma 12A.1)

\[A,B,C \in M_{m \times n}\] \( \implies \)
  1. \[A+B = B+A\]
  2. \[(A+B)+C = A+(B+C)\]
  3. \[\exists \O \in M_{m\times n}, \O A=A\O=A\]
  4. \[\exists {-A} \in M_{m\times n}, A+(-A) = \O\]

Properties of Matrix-Scalar Multiplication (Lemma 12A.2)

\(A,B\in M_{m\times n}\), \(C\in M_{n\times p}\), \(c,d\in\F\)

\( \implies \)
  1. \[cA = Ac\]
  2. \[c(A+B) = cA + cB\]
  3. \[(c+d)A = cA + dA\]
  4. \[c(dA) = (cd)A\]
  5. \[c(AC) = (cA)C = A(cC)\]

Properties of the Transpose (Lemma 12A.3)

\(A,B\in M_{m\times n}\), \(c\in\F\)

\( \implies \)
  1. \[(A+B)^T = A^T + B^T\]
  2. \[(cA)^T = cA^T\]
  3. \[(A^T)^T = A\]

Properties of Matrix Multiplication (Lemma 12A.4)

\(A,G\in M_{m\times n}\), \(B,D\in M_{n\times p}\), \(C\in M_{p\times q}\)

\( \implies \)
  1. \[(A+G)B = AB + GB\]
  2. \[A(B+D) = AB + AD\]
  3. \[(AB)C = A(BC)\]
  4. \[(AB)^T = B^T A^T\]

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)

\(A \in M_{m\times n}\), \(T : \F^n \to \F^m\)

\( \implies \)
  • \([T_A]_S = A\) (13B.3)
  • \(T_{[T]_S} = T\) (13B.4)
  • \(T(\vb x) = [T]_S\vb x\) (13B.7)

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

\( \implies \)
  1. \[(A^T)^{-1} = (A^{-1})^T\]
  2. \((cA)^{-1} = c^{-1}A^{-1}\) for \(c \neq 0\)
  3. \[(AB)^{-1} = B^{-1}A^{-1}\]
  4. \(C,D\in M_{n\times p}\), \(AC=AD \iff C=D\)
  5. \(C\in M_{n\times p}\), \(AC = \O \iff C=\O\)

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)

\[A \in M_{n \times n}\] \( \implies \)
  1. \(\det(A)\) skew-symmetric under row/column interchange
  2. \(\det(A)\) linear on rows/columns