Linear Transformations and Matrix Vector Multiplication
Jul 20, 2024
Linear Transformations and Matrix Vector Multiplication
Introduction
Key Topic: Linear transformations in 2D and their relation to matrices.
Importance: Fundamental to understanding linear algebra; often overlooked initially.
Understanding Linear Transformations
Transformation Defined: A function taking in inputs and providing outputs. Specifically, in linear algebra, it takes vectors as inputs and outputs other vectors.
Visualizing Transformations: Think of moving vectors from their input position to their output position. Use points instead of arrows for simplicity.
Properties of Linear Transformations
Linear Transformation: Special type of transformation maintaining two properties:
All lines remain straight (no curves).
The origin remains fixed.
Examples: Rotations about the origin, scaling, etc.
Numerical Description of Transformations
Basis Vectors: Describing transformations by where the basis vectors (i-hat and j-hat) land.
Vector Decomposition: Any vector v can be written as a linear combination of i-hat and j-hat.
Transformation of Basis Vectors: The output of vector v under transformation can be deduced from where i-hat and j-hat land.
Example Calculation
Given Vector: v with coordinates (-1, 2).
Transformed Basis Vectors:
i-hat maps to (1, -2).
j-hat maps to (3, 0).
Transformed Vector: v lands on (-1*(1, -2) + 2*(3, 0) = (5, 2)).
Formula: Generalize to any vector (x, y).
v lands on (1x + 3y, -2x + 0y).
Matrix Representation: Use a 2x2 matrix with columns representing transformed i-hat and j-hat.
Matrix Vector Multiplication
Interpreting 2x2 Matrix: Columns are transformed versions of basis vectors.
Application: To transform any vector, multiply its coordinates by the matrix.
General Case:
Matrix: [[A, B], [C, D]]
Vector: (x, y)
Result: (Ax + By, Cx + Dy)
Example Transformations
90-degree Rotation:
i-hat: (0, 1)
j-hat: (-1, 0)
Matrix: [[0, -1], [1, 0]]
Shear:
i-hat: (1, 0)
j-hat: (1, 1)
Matrix: [[1, 1], [0, 1]]
Transforming with Given Matrix
Example Matrix: Columns (1, 2) and (3, 1).
Resulting Transformation: Move space such that gridlines stay parallel and evenly spaced.
Linearly Dependent Case: Squishes space onto a line where transformed vectors sit.
Summary
Gridlines and Fixed Origin: Linear transformations preserve parallel gridlines and the origin.
Handful of Numbers: Described by coordinates of basis vectors.
Matrices: Represent these transformations; matrix-vector multiplication computes transformation.
Insight: Matrix as a transformation makes understanding other topics easier (e.g., matrix multiplication, determinants, eigenvalues).