2.3 Matrix representation of linear transformations

Linear transformations are a key concept in abstract algebra, bridging the gap between vector spaces and matrices. This section explores how we can represent these transformations using matrices, allowing us to apply computational techniques to abstract mathematical ideas.

Matrix representation of linear transformations provides a powerful tool for analyzing and manipulating these functions. By converting abstract transformations into concrete matrices, we can leverage matrix algebra to solve problems, compose transformations, and study their properties in various fields of mathematics and science.

Linear transformations and matrices

Matrix representation fundamentals

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• Linear transformations T: V → W between finite-dimensional vector spaces represented by matrix A with respect to chosen bases for V and W
• Matrix representation A dimensions m × n, where n equals dimension of domain V and m equals dimension of codomain W
• Columns of matrix A contain images of basis vectors of V under transformation T, expressed as linear combinations of basis vectors of W
• Matrix representation depends on choice of bases for both domain and codomain vector spaces
• Image T(v) computed by multiplying matrix A by coordinate vector of v with respect to chosen basis of V
• Matrix representation enables application of matrix algebra techniques to study and manipulate linear transformations

Computing matrix representation

• Identify standard bases (or chosen bases) for domain and codomain vector spaces
• Apply linear transformation T to each basis vector of domain space V
• Express resulting vectors (images) as linear combinations of basis vectors of codomain space W
• Coefficients of linear combinations form columns of matrix representation A
• For transformation T: Rn → Rm, resulting matrix A has dimensions m × n
• Verify computed matrix by applying to arbitrary vectors in domain and comparing results with direct application of transformation
• Standard matrix representations for special cases (rotations, reflections, projections in R2 and R3) derived using this process

Matrix representation of linear transformations

Practical computation steps

• Choose bases for domain V and codomain W (often standard bases)
• Apply transformation T to each basis vector ei of V
• Express T(ei) as linear combination of basis vectors in W
• Arrange coefficients as columns of matrix A
• Resulting matrix A has dimensions m × n (m = dim(W), n = dim(V))
• Verify matrix representation by testing on sample vectors
• Example: For T: R2 → R3 defined by T(x, y) = (x + y, x - y, 2x), compute A:
  • T(1, 0) = (1, 1, 2) → first column of A
  • T(0, 1) = (1, -1, 0) → second column of A
  • A = [[1, 1], [1, -1], [2, 0]]

Properties and applications

• Matrix representation independent of specific vectors, depends only on transformation and chosen bases
• Allows conversion of abstract transformations into concrete matrices for computational purposes
• Facilitates analysis of transformation properties (injectivity, surjectivity, invertibility) through matrix properties
• Enables use of matrix algebra for composing transformations and solving related equations
• Useful in various fields (computer graphics, physics, engineering) for representing and manipulating transformations

Linear transformations vs matrix multiplication

Composition and multiplication

• Matrix multiplication corresponds to composition of linear transformations
• For T: V → W and S: W → U with matrix representations A and B, composition S ∘ T has matrix representation BA
• Order of matrix multiplication matches order of function composition: (S ∘ T)(v) = B(Av)
• Identity transformation corresponds to identity matrix
• Invertible linear transformation corresponds to invertible matrix, inverse transformation represented by inverse matrix

Kernel and range correspondence

• Kernel (null space) of linear transformation corresponds to null space of its matrix representation
• Range (image) of linear transformation corresponds to column space of its matrix representation
• Example: For T: R3 → R2 with matrix A = [[1, 2, 3], [4, 5, 6]],
  • Ker(T) = Null(A) = {(x, y, z) | x + 2y + 3z = 0, 4x + 5y + 6z = 0}
  • Range(T) = Col(A) = span{(1, 4), (2, 5), (3, 6)}

Solving problems with matrix representation

Computational techniques

• Compute image of vector v under linear transformation T using matrix multiplication: T(v) = Av
• Solve systems of linear equations from linear transformations using Gaussian elimination or inverse matrices
• Determine kernel and range of linear transformation by analyzing matrix representation
• Compute composition of linear transformations using matrix multiplication
• Analyze properties of linear transformation (invertibility, invariant subspaces) using determinants and eigenvalues of matrix representation

Advanced applications

• Apply change of basis formulas to obtain different matrix representations of same linear transformation with respect to different bases
• Utilize matrix representations to study and classify geometric transformations (rotations, reflections, projections) in various dimensions
• Example: Rotation in R2 by angle θ represented by matrix [[cos θ, -sin θ], [sin θ, cos θ]]
• Use matrix representations to analyze linear transformations in abstract vector spaces (polynomial spaces, function spaces)
• Apply matrix representation techniques to solve differential equations and analyze linear systems in physics and engineering