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.
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
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
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