Glossary
Linear Transformation
A special type of function that takes an input vector and produces an output vector, while preserving vector addition and scalar multiplication.
Example:
Rotating a shape on a coordinate plane is a linear transformation because it maintains the relative positions and scaling of its vertices.
Matrices
Rectangular arrays of numbers, symbols, or expressions arranged in rows and columns, commonly used to represent linear transformations or collections of vectors.
Example:
A 2x2 matrix can represent a transformation that scales an image, making it twice as large.
Matrix Multiplication
A binary operation that combines two matrices to produce a new matrix, where the entries of the result are calculated from the dot product of rows and columns of the input matrices.
Example:
To apply a rotation followed by a scaling to a set of points, you would perform matrix multiplication of the rotation matrix and the scaling matrix.
Preserves Addition
A property of linear transformations where the transformation of a sum of vectors is equal to the sum of the transformations of individual vectors (T(u + v) = T(u) + T(v)).
Example:
If you transform two separate forces acting on an object and then add them, the result is the same as adding the forces first and then transforming their sum; this shows the transformation preserves addition.
Preserves Scalar Multiplication
A property of linear transformations where the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformed vector (T(cu) = cT(u)).
Example:
If you double the velocity vector of a car and then apply a linear transformation, it's equivalent to applying the transformation first and then doubling the transformed velocity vector, demonstrating it preserves scalar multiplication.
Set of n Vectors in R²
Multiple vectors in a two-dimensional space, represented as a 2xn matrix where each column corresponds to an individual vector.
Example:
To track the movement of five different drones simultaneously, you could organize their current positions as a set of n vectors in R² in a 2x5 matrix.
Single Vector in R² (Column Vector)
A vector in a two-dimensional space (R²) typically represented as a 2x1 matrix, or a column with two components (x, y).
Example:
The coordinates of a treasure chest at (5, -3) on a map can be written as a single vector in R², [5; -3].
Transformation Matrix
A specific matrix that uniquely represents a linear transformation, allowing the transformation to be applied to a vector by left-multiplying the vector by this matrix.
Example:
To reflect a point across the y-axis, you would use a specific transformation matrix [[-1, 0], [0, 1]].
Zero Vector (Null Vector)
A vector in which all components are zero, representing the origin in a vector space; linear transformations always map this vector to itself.
Example:
In a game, if a character is at the origin (0,0), its position is represented by the zero vector.