Glossary
Associativity (of composition)
A property indicating that when three or more linear transformations are composed, the grouping of the transformations does not alter the final result.
Example:
If you have transformations A, B, and C, applying A then (B then C) yields the same result as applying (A then B) then C, demonstrating associativity.
Clockwise Rotation
A rotation performed in the same direction as the hands of a clock, which can be achieved by using a negative angle in the standard counterclockwise rotation matrix formula.
Example:
To perform a 45-degree clockwise rotation, you would substitute θ = -45 degrees into the standard rotation matrix.
Composition (of linear transformations)
The process of applying one linear transformation after another, where the output of the first transformation serves as the input for the second.
Example:
First reflecting a shape over the x-axis and then rotating it by 90 degrees is a composition of two linear transformations.
Determinant
A scalar value calculated from the elements of a square matrix, which provides information about the transformation's scaling and orientation change.
Example:
For the matrix [3 1; 2 4], the determinant is (3*4) - (1*2) = 12 - 2 = 10.
Dilation Magnitude
The absolute value of the determinant of a transformation matrix, which quantifies the scaling factor of areas in 2D space under the transformation.
Example:
If a transformation matrix has a determinant of -7, the dilation magnitude is 7, meaning any area will be scaled by a factor of 7.
Identity Matrix
A square matrix with ones on its main diagonal and zeros everywhere else, which acts as a multiplicative identity, leaving any vector unchanged when multiplied.
Example:
The 2x2 identity matrix [1 0; 0 1] will leave any 2D vector exactly as it is when multiplied.
Image of a Vector
The resulting vector obtained after applying a transformation matrix to an original vector.
Example:
If the vector <1, 0> is transformed to <2, 3>, then <2, 3> is the image of a vector <1, 0> under that specific transformation.
Inverse Matrix
A matrix that, when multiplied by the original matrix, results in the identity matrix, effectively reversing the original transformation.
Example:
If matrix A rotates a vector by 60 degrees, its inverse matrix A⁻¹ would rotate it by -60 degrees, bringing it back to its original orientation.
Inverse Transformation
A transformation that completely 'undoes' the effect of another transformation, mapping a transformed vector back to its original state.
Example:
If a transformation scales a vector by a factor of 3, its inverse transformation would scale it back down by a factor of 1/3.
Linear Transformation
A function that moves and stretches vectors in a specific way, ensuring that lines remain straight and the origin stays fixed.
Example:
A linear transformation can rotate a square or stretch it into a rectangle, but it will never bend its sides into curves.
Matrix Multiplication (in context of composition)
The operation used to combine individual transformation matrices into a single matrix that represents the combined effect of multiple linear transformations.
Example:
To find the matrix for a rotation followed by a reflection, you perform matrix multiplication of the reflection matrix by the rotation matrix.
Rotation Matrix
A specific transformation matrix used to rotate a vector counterclockwise by a given angle θ around the origin.
Example:
To rotate a vector by 90 degrees counterclockwise, you would use a rotation matrix with cos(90) and sin(90) values.
Standard Basis
A set of unit vectors (e.g., `<1, 0>` and `<0, 1>` in 2D) whose transformed images form the columns of a transformation matrix.
Example:
Understanding how the standard basis vectors transform is crucial because their images directly populate the columns of the transformation matrix.
Transformation Equations (for rotation)
Algebraic expressions that define the new coordinates (x', y') of a vector after it has been rotated by an angle θ.
Example:
The transformation equations x' = xcos(θ) − ysin(θ) and y' = xsin(θ) + ycos(θ) precisely describe how a point's coordinates change after a rotation.
Transformation Matrix
A special matrix that represents a linear transformation, encoding the rules for how vectors are transformed.
Example:
To reflect a point across the x-axis, you would use a specific transformation matrix like [1 0; 0 -1].
Unit Vectors
Vectors with a magnitude (length) of one, often used as fundamental building blocks or basis vectors in a coordinate system.
Example:
In a 2D Cartesian plane, <1, 0> and <0, 1> are the standard unit vectors that align with the x and y axes, respectively.