Finite Math Review - Transformations, Patterns, Matrices, and Linear Systems

Vocabulary practice flashcards covering basic finite math concepts, including geometric transformations, symmetries, tessellations, frieze notations, golden ratio, matrices, elementary row operations, and determinants.

Pre-Image

The original figure before any transformation occurs, typically labeled with standard letters such as $(A, B, C)$.

Image

The figure produced after a transformation has been applied, labeled using prime notation such as $(A', B', C')$.

Rotation

A transformation that turns a figure around a fixed point called the center of rotation by a specified angle.

Reflection

A transformation that flips a figure over a line, creating a mirror image of the original figure.

Translation

A rigid transformation that slides every point of a figure the exact same distance and direction according to the coordinate rule $(x,y) \rightarrow (x+a, y+b)$.

Dilation

A non-rigid transformation that enlarges or reduces a figure proportionally from a fixed point called the center of dilation using the rule $(x',y') = (a+k(x-a), b+k(y-b))$.

Center of Dilation

The fixed reference point $(a,b)$ from which all points of a geometric figure expand or contract during a dilation.

Scale Factor ($k$)

The ratio of the image coordinate to the original coordinate that dictates whether a dilation is an enlargement ($k > 1$), a reduction ($0 < k < 1$), or identical in size ($k = 1$).

Symmetry

The condition of an object remaining invariant under various geometric transformations.

Center of Rotation

The fixed point around which an object or shape turns when undergoing rotational symmetry or rotation.

Angle of Rotation

The smallest angle through which an object can be turned around its center of rotation so that it coincides with itself.

Order of Symmetry

The total number of times a figure maps onto itself as it rotates through a full rotation from $0^\bullet$ to $360^\bullet$.

Radial Symmetry

Symmetry arranged around a central fixed point, which can be categorized as cyclic or dihedral.

Translational Symmetry

A spatial property where a pattern, design, or shape can be slid in a given direction and match its original layout exactly.

Tessellation

The repeating layout of geometric shapes on a two-dimensional plane without gaps or overlaps, where interior angles around each vertex total $360^\bullet$.

Regular Tessellation

A tessellation constructed using only one type of regular polygon with congruent sides and angles.

Semi-Regular Tessellation

A tessellation formed using two or more types of regular polygons arranged consistently at every vertex.

Irregular Tessellation

A tessellation built with non-regular or irregular shapes that fit together seamlessly with no spaces or overlaps.

Frieze Pattern

A design possessing translational symmetry in one direction, comprising recurring copies of a figure along a straight line.

Hop Notation ($p1$)

A frieze pattern classification characterized solely by translation symmetry.

Step Notation ($p1g$)

A frieze pattern classification featuring translation and glide reflection symmetry.

Slide Notation ($p1m1$)

A frieze pattern classification containing translation and vertical reflection symmetry.

Spinning Hop Notation ($p11g$)

A frieze pattern classification exhibiting translation and $180^\bullet$ rotation symmetry.

Spinning Slide Notation ($p2mg$)

A frieze pattern classification that combines translation, vertical reflection, glide reflection, and $180^\bullet$ rotation symmetry.

Jump Notation ($p11m$)

A frieze pattern classification comprising translation and horizontal reflection symmetry.

Spinning Jump Notation ($p2mm$)

A frieze pattern classification that integrates translation, horizontal reflection, vertical reflection, and $180^\bullet$ rotation symmetry.

Golden Ratio

A proportional relationship between two quantities where the ratio of their sum to the larger quantity equals the ratio of the larger quantity to the smaller quantity, approximately $1.618$.

Golden Rectangle

A rectangle whose ratio of length to width equals approximately $1.618$.

Golden Spiral

A spiral whose growth growth rate is governed by the Golden Ratio.

Golden Angle

The angle formed by dividing a circle according to the Golden Ratio, measuring approximately $137.5^\bullet$.

Fibonacci Sequence

A numerical sequence where every term is generated by summing the two preceding numbers, whose ratio between consecutive terms approaches the Golden Ratio.

Fractal

A self-similar geometric pattern that repeats across multiple scales and dimensions.

Matrix

A rectangular array of numbers or elements arranged into rows and columns with dimensions specified as $m \times n$.

Equal Matrices

Matrices that possess the exact same dimensions ($m \times n$) and identical corresponding entries.

Scalar Multiplication

An operation where every entry of a matrix $A$ is multiplied by a real constant scalar $c$ to yield $cA$.

Matrix Multiplication

An operation combining an $m \times r$ matrix $A$ with an $r \times n$ matrix $B$ to produce an $m \times n$ matrix $AB$ by taking the dot product of rows from $A$ and columns from $B$.

Transpose of a Matrix

A matrix created by interchanging the rows and columns of an original $m \times n$ matrix, resulting in a matrix of size $n \times m$.

Commutative Property of Matrix Addition

The algebraic property stating that matrix addition ordering does not alter the result: $A + B = B + A$.

Associative Property of Matrix Addition

The property stating that matrix grouping during addition does not alter the sum: $(A + B) + C = A + (B + C)$.

Homogeneous Linear Equation

A linear equation in which the constant term on the right-hand side is equal to $0$.

Consistent System

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

Inconsistent System

A system of linear equations that has no solution, represented graphically by parallel lines or non-intersecting planes.

Augmented Matrix

A compact matrix structure containing the coefficients of variables along with the constants of a linear system separated by a vertical line.

Elementary Row Operations (ERO)

Three matrix operations (multiplying a row by a non-zero scalar, swapping rows, or adding a scalar multiple of one row to another) that convert an augmented matrix into simpler form without changing its solution set.

Determinant

A scalar value calculated from a square $n \times n$ matrix, denoted as $\text{det}(A)$ or $|A|$.