1System of Linear Equations (SofLEs)
A collection of one or more linear equations involving the same variables. When evaluated over $\mathbb{Z}_p$, arithmetic is performed modulo $p$.
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Homogeneous System
A system of linear equations where all constant terms are zero, typically written in the form $[A\vert{}\vec{0}]$.
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Rank of a Matrix
The number of nonzero rows in a matrix after it has been reduced to row echelon form.
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Theorem 2.3 (Homogeneous Systems)
If $[A\vert{}\vec{0}]$ is a homogeneous system of $m$ linear equations with $n$ variables where $m < n$, then the system has infinitely many solutions.
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Linear Combination
A vector $\vec{v}$ is a linear combination of vectors $\vec{v}_1, \dots, \vec{v}_k$ if there exist scalars $c_1, \dots, c_k$ such that $\vec{v} = c_1\vec{v}_1 + \dots + c_k\vec{v}_k$.
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Span
The set of all possible linear combinations of a given set of vectors. For example, the span of two non-parallel vectors in $\mathbb{R}^3$ forms a plane through the origin.
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Linearly Dependent
A set of vectors $\{v_1, \dots, v_k\}$ where there are scalars $c_1, \dots, c_k$, at least one of which is not zero, such that $c_1v_1 + \dots + c_kv_k = 0$. This means at least one vector can be written as a linear combination of the others.
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Linearly Independent
A set of vectors that is not linearly dependent; the only way to satisfy $c_1v_1 + \dots + c_kv_k = 0$ is the trivial solution ($c_1 = \dots = c_k = 0$).
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Equivalent Modulo $n$ ($a \equiv b \pmod{n}$)
Integers $a$ and $b$ are equivalent modulo $n$ if $n$ divides $b - a$, which can also be written as $a = b + nk$ for some integer $k$.
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Reduced Form $\pmod{n}$
An integer $a \pmod{n}$ is in reduced form if $0 \le a < n$.
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$\mathbb{Z}_n$
The set of integers modulo $n$, consisting of the elements $\{\overline{0}, \overline{1}, \dots, \overline{n-1}\}$.
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Multiplicative Inverse $\pmod{n}$
Integers $a$ and $b$ are multiplicative inverses modulo $n$ if $ab \equiv 1 \pmod{n}$. An integer $x$ has a multiplicative inverse $\pmod{n}$ if and only if $\gcd(x, n) = 1$.
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Tricolorability of Knots
A property used to distinguish knots by coloring segments with 3 colors. It can be modeled by setting up and solving a homogeneous System of Linear Equations over $\mathbb{Z}_3$.
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Matrix Equality
Two matrices $A$ and $B$ are equal if and only if they have the same dimensions and their corresponding entries are equal ($a_{ij} = b_{ij}$).
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Square Matrix
An $n \times n$ matrix where the number of rows equals the number of columns.
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Diagonal Matrix
A square matrix whose non-diagonal entries are all zero.
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Identity Matrix ($I_n$)
A square $n \times n$ diagonal matrix with $1$s in all of its diagonal entries and $0$s everywhere else.
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Zero Matrix ($\mathcal{O}$)
A matrix in which all entries are zero. It acts as the additive identity, meaning $A + \mathcal{O} = A$, and $A \mathcal{O} = \mathcal{O}$.
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Matrix Transpose ($A^T$)
The matrix formed by swapping the rows and columns of $A$. The $ij$-th entry of $A^T$ is the $ji$-th entry of $A$.
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Matrix Multiplication Dimension Rule
If $A$ is an $m \times n$ matrix and $B$ is an $n \times r$ matrix, the product $AB$ is defined and results in an $m \times r$ matrix.
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Commutativity of Matrix Addition
For any two matrices $A$ and $B$ of the same size, $A + B = B + A$.
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Non-Commutativity of Matrix Multiplication
For matrices $A$ and $B$, the product $AB$ does not necessarily equal $BA$. Matrix multiplication is generally not commutative.
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Transpose of a Transpose
$(A^T)^T = A$.
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Transpose of a Sum
$(A + B)^T = A^T + B^T$.
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Transpose of a Scalar Multiple
$(kA)^T = k(A^T)$.
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Transpose of a Product
$(AB)^T = B^T A^T$.
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Identity Matrix Multiplication Property
For any $m \times n$ matrix $A$, multiplication by the appropriately sized identity matrix leaves it unchanged: $I_m A = A = A I_n$.
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Invertible Matrix
An $n \times n$ matrix $A$ is invertible if there exists an $n \times n$ matrix $A^{-1}$ such that $A A^{-1} = I_n$ and $A^{-1} A = I_n$.
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Uniqueness of Matrix Inverse
If an $n \times n$ matrix $A$ is invertible, its inverse $A^{-1}$ is unique.
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Inverse of a $2 \times 2$ Matrix
If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $A$ is invertible if $ad - bc \neq 0$, and $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
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Solving Linear Systems with Inverses
If $A$ is an invertible $n \times n$ matrix, the system $Ax = b$ has a unique solution given by $x = A^{-1}b$ for any $b \in \mathbb{R}^n$.
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Inverse of an Inverse
If $A$ is invertible, then $A^{-1}$ is also invertible, and $(A^{-1})^{-1} = A$.
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Inverse of a Matrix Product
If $A$ and $B$ are $n \times n$ invertible matrices, then their product $AB$ is also invertible, and $(AB)^{-1} = B^{-1}A^{-1}$.
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Algorithm for Finding $A^{-1}$
Construct the augmented block matrix $[A \vert{} I_n]$. Perform row operations to reduce $A$ to the identity matrix. If successful, the resulting matrix is $[I_n \vert{} A^{-1}]$.
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Fundamental Theorem of Invertible Matrices (Part 1)
An $n \times n$ matrix $A$ is invertible if and only if the system $Ax = b$ has a unique solution for every $b \in \mathbb{R}^n$.
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Fundamental Theorem of Invertible Matrices (Part 2)
An $n \times n$ matrix $A$ is invertible if and only if the homogeneous system $Ax = 0$ has only the trivial solution.
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Fundamental Theorem of Invertible Matrices (Part 3)
An $n \times n$ matrix $A$ is invertible if and only if the reduced row echelon form (RREF) of $A$ is the identity matrix $I_n$.
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Trivial Solution
The solution to a homogeneous equation $Ax = 0$ where all variables are zero ($x = 0$). If vectors are linearly independent, this is the only solution to their linear combination equating to zero.