Linear Systems, Matrix Algebra, and Analytic Geometry

Comprehensive vocabulary flashcards covering systems of linear equations, row reduction techniques, spanning sets, linear independence, matrix algebra operations, transposes, orthogonal projections, and analytic geometry of lines and planes in Euclidean space.

Linear Equation

An equation in $n$ variables that can be written in the form $a_1x_1 + a_2x_2 + \dots + a_nx_n = b$, where the coefficients $a_1, \dots, a_n$ and the constant term $b$ are constants.

System of Linear Equations

A finite set of linear equations involving the same variables. A solution is a vector that simultaneously satisfies every equation in the system.

Consistent System

A linear system that possesses at least one solution (either exactly one unique solution or infinitely many solutions).

Inconsistent System

A linear system that has no solution.

Equivalent Systems

Two or more systems of equations that have the exact same solution set.

Augmented Matrix

A matrix written in the form $[A \mid \mathbf{b}]$ that incorporates both the coefficient matrix $A$ and the constant column vector $\mathbf{b}$ of a linear system.

Row Echelon Form (REF)

A matrix structure where all zero rows are grouped at the bottom, and in each nonzero row, the first nonzero entry (leading entry) lies strictly to the right of the leading entry in the row above it.

Elementary Row Operations

Three operations that produce an equivalent matrix/system: 1. Interchange two rows ($R_i \leftrightarrow R_j$); 2. Multiply a row by a nonzero scalar ($R_i \leftarrow kR_i, k \neq 0$); 3. Add a scalar multiple of one row to another row ($R_i \leftarrow R_i + kR_j$).

Reduced Row Echelon Form (RREF)

A matrix form that satisfies all REF conditions, has a leading entry equal to $1$ (a leading 1) in every nonzero row, and contains zeros everywhere else above and below each leading 1 in its column.

Gaussian Elimination

The procedure of applying elementary row operations to reduce an augmented matrix to Row Echelon Form (REF), followed by solving for variables via back substitution.

Gauss–Jordan Elimination

The procedure of applying elementary row operations to reduce an augmented matrix all the way to Reduced Row Echelon Form (RREF) to read off the solutions directly.

Rank of a Matrix

The number of leading entries (pivots) present in a matrix $A$ after it has been reduced to Row Echelon Form, denoted $\text{rank}(A)$.

Rank Theorem (Version 1)

For the coefficient matrix $A$ of a consistent linear system with $n$ variables: $\text{number of parameters} = n - \text{rank}(A)$.

Homogeneous System

A linear system in which every constant term is zero, expressed in matrix form as $A\mathbf{x} = \mathbf{0}$. It is always consistent because it possesses at least the trivial solution $\mathbf{x} = \mathbf{0}$.

Homogeneous System with Fewer Equations than Variables ($m < n$)

A homogeneous system $[A \mid \mathbf{0}]$ with $m$ equations and $n$ variables where $m < n$, which is guaranteed to have infinitely many solutions because $\text{rank}(A) \le m < n$, leaving at least one free parameter.

Span

The set of all possible linear combinations formed from a set of vectors $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\} \subseteq \mathbb{R}^n$, defined as $\text{span}(S) = \{c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k : c_1, \dots, c_k \in \mathbb{R}\}$.

Spanning Set for $\mathbb{R}^n$

A set of vectors $S \subseteq \mathbb{R}^n$ whose span generates the entire space, such that $\text{span}(S) = \mathbb{R}^n$.

Consistency Test for a Linear Combination

The principle stating that a system $A\mathbf{c} = \mathbf{b}$ is consistent if and only if the vector $\mathbf{b}$ can be written as a linear combination of the columns of matrix $A$.

Linear Independence

A property of a vector set $S = \{\mathbf{v}_1, \dots, \mathbf{v}_k\} \subseteq \mathbb{R}^n$ where the vector equation $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ forces all scalar coefficients to be zero: $c_1 = c_2 = \dots = c_k = 0$.

Linear Dependence

A property of a vector set where there exists a linear combination $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ with at least one nonzero scalar coefficient $c_i \neq 0$, meaning at least one vector can be built from the others.

Homogeneous-System Test for Linear Independence

A test where vectors are arranged as columns of matrix $A = [\mathbf{v}_1\ \mathbf{v}_2\ \cdots\ \mathbf{v}_k]$; the vectors are linearly independent if and only if $A\mathbf{c} = \mathbf{0}$ has only the trivial solution $\mathbf{c} = \mathbf{0}$.

Row-Vector Rank Test

A criterion where $k$ row vectors in $\mathbb{R}^n$ placed into the rows of a $k \times n$ matrix $A$ are linearly independent if and only if $\text{rank}(A) = k$.

Quick Dimension Test

The rule establishing that any set of $k$ vectors in $\mathbb{R}^n$ is guaranteed to be linearly dependent whenever $k > n$.

Matrix

A rectangular array of numbers with $m$ rows and $n$ columns ($m \times n$), where $a_{ij}$ denotes the entry in row $i$ and column $j$.

Main Diagonal

The collection of entries $a_{11}, a_{22}, \dots$ in a matrix where the row index and column index are equal.

Diagonal Matrix

A square matrix in which every entry outside the main diagonal (all off-diagonal entries) is equal to $0$.

Scalar Matrix

A diagonal matrix that has the exact same scalar value on every main diagonal entry, represented in the form $cI_n$.

Identity Matrix ($I_n$)

A square $n \times n$ matrix with $1$ on every position along the main diagonal and $0$ in every other position.

Matrix Product

For an $m \times k$ matrix $A$ and a $k \times r$ matrix $B$, the resulting $m \times r$ matrix $AB$ whose $(i, j)$ entry is given by $(AB)_{ij} = \sum_{\ell=1}^k a_{i\ell}b_{\ell j}$, computed as the dot product of row $i$ of $A$ with column $j$ of $B$.

Non-Commutativity of Matrix Multiplication

The property that matrix products generally depend on order, such that $AB \neq BA$ in general even when both dimensions permit multiplication; furthermore, $AB = O$ does not imply $A = O$ or $B = O$.

Matrix Powers

An operation defined for square $n \times n$ matrices and integers $k \ge 0$, where $A^0 = I_n$, $A^1 = A$, and $A^k = AA\cdots A$ ($k$ factors), satisfying the exponent law $A^r A^s = A^{r+s}$.

Transpose of a Matrix

For an $m \times n$ matrix $A = [a_{ij}]$, the $n \times m$ matrix $A^T$ formed by swapping rows and columns such that $(A^T)_{ij} = a_{ji}$.

Transpose of a Matrix Product

The algebraic identity stating that transposing a matrix product reverses the order of multiplication: $(AB)^T = B^T A^T$, and more generally $(ABC)^T = C^T B^T A^T$.

Row and Column Selection Rule

Given an $m \times n$ matrix $A$ and a standard basis vector $\vec{e}_j$, the product $\vec{e}_j^T A$ selects row $j$ of $A$ (with $\vec{e}_j \in \mathbb{R}^m$), while $A\vec{e}_j$ selects column $j$ of $A$ (with $\vec{e}_j \in \mathbb{R}^n$).

Two-Dimensional Plane in $\mathbb{R}^n$

A geometric object described by the equation $\mathbf{x} = \mathbf{p} + s\mathbf{u} + t\mathbf{v}$ for parameters $s, t \in \mathbb{R}$, where $\mathbf{p}$ is a position vector and $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ are nonzero, nonparallel direction vectors.

Orthogonal Projection onto a Vector

For a nonzero vector $\mathbf{w}$, the orthogonal projection of vector $\mathbf{v}$ onto $\mathbf{w}$ is $\text{proj}_{\mathbf{w}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}\mathbf{w}$, representing the component parallel to $\mathbf{w}$, while $\mathbf{v} - \text{proj}_{\mathbf{w}}(\mathbf{v})$ represents the perpendicular remainder.

Closest Point and Distance from a Point to a Line

Given a line $\ell$ passing through point $A$ with direction $\mathbf{d}$ and an external point $B$, the closest point on $\ell$ is $H = A + \text{proj}_{\mathbf{d}}(\vec{AB})$ and the distance is $\text{dist}(B, \ell) = \|\vec{AB} - \text{proj}_{\mathbf{d}}(\vec{AB})\|$.

Closest Point and Distance from a Point to a Plane

Given a plane $\mathcal{P}$ containing point $A$ with normal vector $\mathbf{n} \neq \mathbf{0}$ and an external point $B$, the perpendicular displacement is $\vec{HB} = \text{proj}_{\mathbf{n}}(\vec{AB})$, yielding distance $\text{dist}(B, \mathcal{P}) = \|\text{proj}_{\mathbf{n}}(\vec{AB})\|$ and closest point $H = B - \text{proj}_{\mathbf{n}}(\vec{AB})$.

Intersecting Lines in $\mathbb{R}^3$

A configuration of two lines that possess nonparallel direction vectors and share exactly one common intersection point.

Parallel and Distinct Lines in $\mathbb{R}^3$

A configuration of two lines whose direction vectors are scalar multiples of each other, but which share no points in common.

Coincident Lines in $\mathbb{R}^3$

A configuration where two vector equations describe the exact same geometric line; their direction vectors are scalar multiples and every point is shared.

Skew Lines in $\mathbb{R}^3$

Two lines in three-dimensional space that are nonparallel (direction vectors are not scalar multiples) and have no common intersection point, meaning they do not lie within a single common plane.