| Term | Definition |
|---|---|
| column | The vertical lines of elements in a matrix. |
| dot product | A scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them; equals zero when vectors are perpendicular. |
| matrix | A rectangular array of numbers arranged in rows and columns that represents a linear transformation. |
| matrix product | The product of two matrices that represents the composition of their corresponding linear transformations. |
| row | The horizontal lines of elements in a matrix. |
| Term | Definition |
|---|---|
| domain | The set of all possible input values for which a function is defined. |
| parameter | An independent variable (often denoted t) used to express the coordinates of points on a curve in parametric form. |
| parametric equations | A pair of equations that express x and y coordinates as functions of a parameter, typically written as x(t) and y(t). |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| Term | Definition |
|---|---|
| 2 × 2 matrix | A square matrix with 2 rows and 2 columns. |
| column vector | Vectors represented as columns in a matrix; when two column vectors form a 2×2 matrix, the absolute value of the determinant gives the area of the parallelogram they span. |
| determinant | A scalar value calculated from a square matrix that determines whether the matrix is invertible; for a 2×2 matrix [a b; c d], the determinant equals ad - bc. |
| identity matrix | A square matrix with 1s on the main diagonal (from top left to bottom right) and 0s everywhere else. |
| invertibility | The property of a square matrix that has an inverse; a matrix is invertible if and only if its determinant is nonzero. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| parallel vectors | Vectors that have the same or opposite direction, resulting from scalar multiplication of a vector by a constant. |
| parallelogram | A quadrilateral formed by two vectors; the area of the parallelogram spanned by two vectors equals the absolute value of the determinant of the matrix formed by those vectors. |
| row vector | Vectors represented as rows in a matrix; when two row vectors form a 2×2 matrix, the absolute value of the determinant gives the area of the parallelogram they span. |
| square matrix | A matrix with the same number of rows and columns; only square matrices can have determinants and inverses. |
| Term | Definition |
|---|---|
| input vector | The vector that is mapped or transformed by a linear transformation. |
| linear transformation | A function that maps vectors to vectors while preserving vector addition and scalar multiplication, represented by a matrix. |
| matrix multiplication | The operation of multiplying a transformation matrix by a vector or matrix to produce output vectors. |
| output vector | The resulting vector produced by applying a linear transformation to an input vector. |
| ℝ² | The two-dimensional real vector space consisting of all ordered pairs of real numbers. |
| transformation matrix | A 2 × 2 matrix A that represents a linear transformation, where L(v) = Av for vectors v in ℝ². |
| zero vector | A vector with components ⟨0, 0⟩ that occurs when the tail and head are at the same point. |
| Term | Definition |
|---|---|
| composition of functions | A function operation where one function is applied to the output of another function, written as (f ∘ g)(x) = f(g(x)). |
| composition of linear transformations | The result of applying one linear transformation followed by another linear transformation. |
| determinant | A scalar value calculated from a square matrix that determines whether the matrix is invertible; for a 2×2 matrix [a b; c d], the determinant equals ad - bc. |
| dilation | A linear transformation that scales regions by a constant factor, with the magnitude determined by the absolute value of the determinant. |
| inverse transformations | Two linear transformations that are inverses if their composition maps any vector to itself, effectively undoing each other's effects. |
| linear transformation | A function that maps vectors to vectors while preserving vector addition and scalar multiplication, represented by a matrix. |
| matrix | A rectangular array of numbers arranged in rows and columns that represents a linear transformation. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| matrix product | The product of two matrices that represents the composition of their corresponding linear transformations. |
| rotation | A linear transformation that rotates every vector by a fixed angle about the origin without changing its length. |
| unit vector | A vector with a magnitude of 1, often used to indicate direction. |
| vector | A mathematical object with both magnitude and direction, represented as an ordered pair of components in ℝ². |
| Term | Definition |
|---|---|
| discrete intervals | Separate, distinct time periods or steps used to measure changes in a system, rather than continuous time. |
| future states | The predicted conditions or distributions of a system at subsequent time steps using matrix multiplication. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| matrix models | Mathematical representations using matrices to represent transitions or changes between different states in a system. |
| past states | The predicted conditions or distributions of a system at previous time steps using the inverse of a transition matrix. |
| percent change | The relative change in a quantity expressed as a percentage, which is related to the growth factor in an exponential model. |
| repeated multiplication | A process where an initial value is multiplied by the same proportion multiple times, which can be modeled using logarithmic functions. |
| state vector | A column vector that represents the distribution or values across different states at a particular point in time. |
| steady state | A distribution between states that remains unchanged from one step to the next after repeated matrix multiplication. |
| transition matrix | A matrix that models the probabilities or rates of moving from one state to another in a system. |
| transitions between states | Changes or movements from one condition or situation to another in a system being modeled. |
| Term | Definition |
|---|---|
| horizontal extrema | The maximum and minimum x-coordinates reached by a particle during its motion, found by identifying extrema of x(t). |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| particle motion | The path and position of a particle as it moves through space over time, modeled using parametric equations. |
| planar motion | The movement of a particle or object in a two-dimensional plane. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| vertical extrema | The maximum and minimum y-coordinates reached by a particle during its motion, found by identifying extrema of y(t). |
| x-intercept | The point where a graph crosses or touches the x-axis, occurring at (a, 0) when a is a real zero of the function. |
| y-intercepts | The points where the particle's path crosses the y-axis, corresponding to the real zeros of x(t). |
| Term | Definition |
|---|---|
| average rate of change | The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b]. |
| direction of motion | The path direction a particle follows in the plane, determined by whether x(t) and y(t) are increasing or decreasing. |
| parametric planar motion function | A function that describes the motion of a particle in a plane using a parameter (typically time) to define both x and y coordinates independently. |
| parametrization | A representation of a curve using a pair of functions (x(t), y(t)) where both x and y are expressed in terms of a parameter t. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| slope of the graph | The ratio of the average rate of change of y to the average rate of change of x between two points on a parametric curve. |
| Term | Definition |
|---|---|
| circular path | A curve traced in the plane that forms a circle, defined by parametric equations. |
| counterclockwise revolution | Motion around a circle in the counterclockwise direction, completing a full 360-degree rotation. |
| line segment | The portion of a line between two endpoints, characterized by a starting point and an ending point. |
| linear path | A straight line segment connecting two points in the coordinate plane. |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| parametrically | Expressed using parametric equations where x and y coordinates are defined as functions of a parameter, typically time (t). |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| transformation | Changes applied to a parent function such as translations, reflections, stretches, or compressions. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| equation involving two variables | A mathematical statement with an equals sign containing two different variables, which can describe one or more functions or be graphed in a coordinate plane. |
| implicitly defined function | A function defined by an equation relating x and y, rather than explicitly solving for y in terms of x. |
| ordered pairs | Points on a graph represented as (x, y) coordinates that satisfy the equation of a function. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| solutions to an equation | The ordered pairs of values that satisfy an equation involving two variables and can be plotted as points on a graph. |
| Term | Definition |
|---|---|
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| circle | A special case of an ellipse where the horizontal and vertical radii are equal (a = b). |
| conic sections | Curves formed by the intersection of a plane with a cone, including parabolas, ellipses, circles, and hyperbolas. |
| ellipse | A conic section formed when a plane intersects a cone at an angle, or the set of points where the sum of distances to two foci is constant. |
| horizontal radius | The distance from the center of an ellipse to its edge along the horizontal axis, represented by the value a. |
| hyperbola | A conic section formed when a plane intersects both nappes of a cone, or the set of points where the difference of distances to two foci is constant. |
| parabola | A conic section formed when a plane intersects a cone parallel to its side, or the set of points equidistant from a focus and directrix. |
| vertex | The point (h, k) that represents the center or turning point of a parabola. |
| vertical radius | The distance from the center of an ellipse to its edge along the vertical axis, represented by the value b. |
| Term | Definition |
|---|---|
| conic sections | Curves formed by the intersection of a plane with a cone, including parabolas, ellipses, circles, and hyperbolas. |
| domain | The set of all possible input values for which a function is defined. |
| ellipse | A conic section formed when a plane intersects a cone at an angle, or the set of points where the sum of distances to two foci is constant. |
| hyperbola | A conic section formed when a plane intersects both nappes of a cone, or the set of points where the difference of distances to two foci is constant. |
| implicitly defined function | A function defined by an equation relating x and y, rather than explicitly solving for y in terms of x. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| invertible function | A function that has an inverse function; a one-to-one function where each output corresponds to exactly one input. |
| parabola | A conic section formed when a plane intersects a cone parallel to its side, or the set of points equidistant from a focus and directrix. |
| parameter | An independent variable (often denoted t) used to express the coordinates of points on a curve in parametric form. |
| parametric equations | A pair of equations that express x and y coordinates as functions of a parameter, typically written as x(t) and y(t). |
| parametrization | A representation of a curve using a pair of functions (x(t), y(t)) where both x and y are expressed in terms of a parameter t. |
| trigonometric parametrization | A method of representing curves using trigonometric functions (such as sine, cosine, secant, and tangent) as the parametric equations. |
| Term | Definition |
|---|---|
| angle measure | The input value in a polar function that represents the direction from the positive x-axis, typically measured in radians or degrees. |
| components | The horizontal (a) and vertical (b) values of a vector, where a = x₂ - x₁ and b = y₂ - y₁. |
| directed line segment | A line segment with a specified direction from a starting point to an ending point. |
| direction | The orientation of a vector, which is parallel to the line segment from the origin to the point with coordinates (a, b). |
| dot product | A scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them; equals zero when vectors are perpendicular. |
| head | The ending point or tip of a vector. |
| Law of Cosines | A relationship used to find side lengths or angle measures in a triangle when given other side lengths and angles. |
| Law of Sines | A relationship stating that the ratio of a side length to the sine of its opposite angle is constant for all sides and angles in a triangle. |
| magnitude | The length of a vector, calculated as the square root of the sum of the squares of its components. |
| parallel vectors | Vectors that have the same or opposite direction, resulting from scalar multiplication of a vector by a constant. |
| perpendicular | Two vectors are perpendicular when the angle between them is 90 degrees, indicated by a dot product of zero. |
| scalar multiplication | The multiplication of a constant (scalar) by a vector, resulting in a new vector whose components are each multiplied by that constant. |
| standard basis vectors | The unit vectors →i = ⟨1, 0⟩ and →j = ⟨0, 1⟩ that point in the positive x and y directions, respectively, in ℝ². |
| tail | The starting point or beginning of a vector. |
| unit vector | A vector with a magnitude of 1, often used to indicate direction. |
| vector | A mathematical object with both magnitude and direction, represented as an ordered pair of components in ℝ². |
| vector addition | The process of combining two or more vectors to produce a resultant vector. |
| vector sum | The addition of two vectors by adding their corresponding components to produce a new vector. |
| zero vector | A vector with components ⟨0, 0⟩ that occurs when the tail and head are at the same point. |
| Term | Definition |
|---|---|
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| planar motion | The movement of a particle or object in a two-dimensional plane. |
| position vector | A vector that represents the location of a particle relative to the origin, with magnitude equal to the distance from the origin. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving regardless of direction. |
| vector-valued function | A function that outputs vectors, typically expressed as p(t) = ⟨x(t), y(t)⟩ or p(t) = x(t)i + y(t)j, where each input t produces a vector output. |
| velocity vector | A vector-valued function v(t) = ⟨x'(t), y'(t)⟩ that represents the rate of change of position with respect to time, indicating both direction and speed of motion. |