4.2 Matrix representations of linear transformations

Linear transformations are mathematical functions that map vectors between spaces while preserving addition and scalar multiplication. Matrices provide a powerful way to represent these transformations, encoding their effects on basis vectors and allowing for efficient computations.

Matrix representations simplify complex transformations into multiplication operations. By applying the transformation to basis vectors and recording the results as matrix columns, we can easily perform and analyze linear transformations using familiar matrix operations.

Linear transformations with matrices

Defining linear transformations

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• Linear transformations map vectors between vector spaces while preserving vector addition and scalar multiplication
• Matrices represent linear transformations by encoding effects on basis vectors of input space
• Columns of matrix representation correspond to images of basis vectors under linear transformation
• For linear transformation T: R^n → R^m, matrix representation becomes m × n matrix
• Matrix representation depends on choice of basis for domain and codomain vector spaces
• Standard matrix representations use standard basis vectors for input and output spaces
• Process of finding matrix representation involves applying transformation to each basis vector and recording results as matrix columns

Matrix representation process

• Apply linear transformation to each basis vector of input space
• Express resulting vectors as linear combinations of basis vectors in output space
• Coefficients of linear combinations form columns of matrix representation
• Ensure number of matrix rows matches output space dimension, columns match input space dimension
• Use change of basis formula to convert between different matrix representations for non-standard bases
• Verify matrix representation by testing on arbitrary vectors and comparing with direct application of linear transformation
• Special cases like rotations, reflections, and scaling have characteristic matrix representations
  • Derive using geometric intuition or trigonometric formulas
  • Example: 2D rotation by angle θ has matrix representation $\begin{bmatrix} \cos θ & -\sin θ \\ \sin θ & \cos θ \end{bmatrix}$
  • Example: Reflection about y-axis has matrix representation $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

Matrix representation of linear transformations

Performing linear transformations

• Action of linear transformation on vector equates to multiplying matrix representation by vector
• For transformation T represented by matrix A and vector v, transformed vector becomes T(v) = Av
• Matrix and vector dimensions must be compatible for multiplication
  • m × n matrix multiplied by n × 1 vector results in m × 1 vector
• Matrix multiplication performed row by column, each entry in resulting vector becomes dot product
• Order of multiplication matters: vector must be on right side of matrix (post-multiplication)
• Composite transformations represented by matrix multiplication of respective matrices
• Associative property of matrix multiplication allows efficient computation of multiple transformations
  • Example: For transformations A, B, C and vector v, (ABC)v = A(B(Cv))

Analyzing transformation properties

• Rank of matrix representation equals dimension of range (image) of linear transformation
• Nullity of matrix representation equals dimension of kernel (null space) of linear transformation
• Determinant of square matrix representation indicates invertibility of transformation
  • Non-zero determinant: invertible transformation
  • Zero determinant: non-invertible transformation
• Eigenvalues and eigenvectors of matrix representation provide information about invariant subspaces and scaling factors
• Trace of matrix representation remains invariant under similarity transformations and relates to sum of eigenvalues
• Singular value decomposition (SVD) of matrix representation reveals principal directions and magnitudes of stretching or compression
• Condition number of matrix representation indicates sensitivity of linear transformation to small input changes
  • Crucial for numerical stability in computations
  • Example: High condition number suggests transformation amplifies small errors, leading to unstable computations

Linear transformations via matrix multiplication

Matrix multiplication mechanics

• Multiply matrix representation by vector to perform linear transformation
• Ensure matrix and vector dimensions are compatible
  • m × n matrix multiplied by n × 1 vector yields m × 1 vector
• Perform row-by-column multiplication
  • Each entry in resulting vector becomes dot product of matrix row and vector
• Maintain correct order: matrix on left, vector on right (post-multiplication)
• Example: For 2×2 matrix A and 2×1 vector v, $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, v = \begin{bmatrix} x \\ y \end{bmatrix}$
  • Transformed vector: $Av = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$

Composite transformations

• Represent composite transformations through matrix multiplication of individual transformation matrices
• Order of multiplication matters: rightmost matrix applies first, leftmost matrix applies last
• Associative property enables efficient computation of multiple transformations
• Example: For transformations A, B, C applied in order (C first, then B, then A):
  • Composite transformation matrix: ABC
  • Applied to vector v: (ABC)v = A(B(Cv))
• Use matrix multiplication to combine multiple transformations into a single matrix
  • Reduces computational complexity when applying to multiple vectors

Properties of linear transformations using matrices

Rank and nullity

• Rank of matrix representation equals dimension of transformation's range (image)
  • Determines number of linearly independent output vectors
• Nullity of matrix representation equals dimension of transformation's kernel (null space)
  • Represents number of linearly independent vectors mapped to zero vector
• Rank-nullity theorem: rank(A) + nullity(A) = number of columns in A
  • Fundamental relationship between input and output spaces
• Example: For a 3×3 matrix with rank 2, nullity must be 1 (2 + 1 = 3)

Determinant and invertibility

• Determinant of square matrix representation indicates transformation's invertibility
  • Non-zero determinant: invertible (one-to-one and onto) transformation
  • Zero determinant: non-invertible transformation (many-to-one or not onto)
• Determinant also represents scaling factor of transformation on volumes
  • Absolute value of determinant gives factor by which volumes are scaled
• Example: Determinant of rotation matrix always equals 1, preserving areas and volumes

Eigenvalues and eigenvectors

• Eigenvalues of matrix representation represent scaling factors along specific directions
• Eigenvectors correspond to directions unchanged by transformation (except for scaling)
• Characteristic equation det(A - λI) = 0 used to find eigenvalues
• Eigenvectors found by solving (A - λI)v = 0 for each eigenvalue λ
• Example: For scaling matrix $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$
  • Eigenvalues: 2 and 3
  • Eigenvectors: [1, 0]^T and [0, 1]^T respectively