## 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)

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

$\vb u \cp \vb v = \mqty(u_2 v_3 - u_3 v_2 \\ -{(u_1 v_3 - u_3 v_1)} \\ u_1 v_2 - u_2 v_1) = \det(\mqty(\vb i&\vb j&\vb k\\u_1&u_2&u_3\\v_1&v_2&v_3))$

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

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)

$$A \in M_{n \times n}$$, $$J \leq n$$

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

$A \in M_{n \times n}$

$$(\adj(A))_{ij} = C_{ji}(A)$$ (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)

$V \subseteq \F^n$
1. $\vb 0 \in V$
2. $$\forall \vb x,\vb y \in V$$, $$\vb x + \vb y \in V$$
3. $$\forall \vb x \in V$$ and $$c \in \F$$, $$c\vb x \in V$$

Trivial Subspace (Example 17A.1)

$$V = \{\vb 0\}$$ and $$V = \F^n$$

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 $$V$$

1. $B \subseteq V$
2. $\Span(B) = V$
3. $$B$$ is linearly independent

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)

• Set $$V$$ of “vectors”
• “Addition” $$\oplus:V^2\to V$$
• Field $$(\F, +, \times)$$ of “scalars”
• “Multiplication” $$\odot:V\times\F\to V$$

For all $$\vb v,\vb w,\vb z \in V$$ and $$c,d\in\F$$, there is closure:

1. $\vb v \oplus \vb w \in V$
2. $c \odot \vb v \in V$

and eight axioms hold:

1. $$\vb v \oplus \vb w = \vb w \oplus \vb v$$ (additive commutativity)
2. $$(\vb v \oplus \vb w) \oplus \vb z = \vb v \oplus \vb w \oplus \vb z$$ (additive associativity)
3. $$\exists \vb 0 \in V$$ where $$\vb v \oplus \vb 0 = \vb v$$ (additive identity)
4. $$\exists (-\vb v) \in V$$ where $$\vb v \oplus (-\vb v) = \vb 0$$ (additive inverse)
5. $$c \odot (\vb v \oplus \vb w) = (c \odot \vb v) \oplus (c \odot \vb w)$$ (vector distributivity)
6. $$(c+d)\odot \vb v = (c\odot \vb v) + (d\odot \vb v)$$ (scalar distributivity)
7. $$(c\times d)\odot \vb v = c \odot (d\odot \vb v)$$ (multiplicative compatibility)
8. $$1 \odot \vb v = \vb v$$ (multiplicative identity)

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)

$z,w\in\C$ $$\implies$$
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

EROs and the Determinant (Corollary 15B.4)

$$A \in M_{n \times n}$$ and $$B$$ is $$A$$ after an ERO of type

$$\implies$$
1. $\det(B) = -\det(A)$
2. $\det(B) = m\det(A)$
3. $\det(B) = \det(A)$

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)

$A \in \Mnn$ $$\implies$$
1. $\Delta_A(t) = b_0 + b_1 t + \dotsb + b_n t^n$
2. $b_n = (-1)^n$
3. $b_{n-1} = (-1)^{n-1}\tr(A)$
4. $b_0 = \det(A)$

Properties of the Complex $$\Delta_A$$ (Lemma 16B.4)

$A \in \Mnn(\C)$ $$\implies$$
1. $\sum_{i=1}^n \lambda_i = \tr(A) = (-1)^{n-1} b_{n-1}$
2. $\prod_{i=1}^n \lambda_i = \det(A) = b_0$

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

$$\iff$$
1. $V \neq \varnothing$
2. $$\forall \vb x,\vb y \in V$$ and $$c \in \F$$, $$c\vb x + \vb y \in V$$

Example 17A.1

• $$\Span(S)$$ is a subspace of $$\F^n$$ for any non-empty $$S \subseteq \F^n$$
• $$\Col(A)$$ is a subspace of $$\F^m$$ for any $$A \in M_{m \times n}$$
• Solution set to $$A\vb x = \vb 0$$ is a subspace of $$\F^n$$
• $$E_\lambda$$ is a subspace of $$\F^n$$ if $$\lambda$$ is an eigenvalue of $$A$$
• $$R(T)$$ is a subspace of $$\F^m$$ for $$T : \F^n \to \F^m$$
• $$N(T)$$ is a subspace of $$\F^n$$

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

• $S = \{\vb v_1,\dotsc,\vb v_p\}$
• $A = (\vb v_1,\dotsc,\vb v_p)$
• $$U$$ the set of $$A$$’s pivot columns
$$\implies$$
• $$S$$ linearly independent $$\iff$$ $$\rank(A) = p$$
• $$U$$ linearly independent
• $$U \subsetneq V \subseteq S$$ linearly dependent
• $\Span(U) = \Span(S)$

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 $$V$$

 $T(\vb x) = {}_{B_2}[T]_{B_1}\,[\vb x]_{B_1}$