4.1 Introduction to linear transformations

Linear transformations are functions that preserve vector operations between vector spaces. They're the backbone of this chapter, showing how we can map vectors while keeping their structure intact. Think of them as special functions that play nice with vectors.

These transformations have two key properties: additivity and homogeneity. They're represented by matrices, which make calculations easier. We'll explore concepts like kernel, range, rank, and nullity to understand how these transformations behave and what they reveal about vector spaces.

Linear transformations

Definition and key properties

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• Linear transformation functions T: V → W between vector spaces preserve vector addition and scalar multiplication
• Two key properties define linear transformations
  • Additivity preserves vector addition: T(u + v) = T(u) + T(v) for all vectors u and v
  • Homogeneity preserves scalar multiplication: T(cv) = cT(v) for all scalars c and vectors v
• Matrices represent linear transformations with columns showing images of standard basis vectors
• Kernel (null space) contains all vectors mapping to the zero vector in codomain
• Range (image) includes all vectors in codomain obtained by applying transformation to domain vectors
• Rank measures dimension of range while nullity measures dimension of kernel
• Rank-nullity theorem states rank(T) + nullity(T) = dim(V) for linear transformation T: V → W

Important concepts and theorems

• Linear transformations preserve vector operations and structure
• Matrix representation simplifies calculations and analysis
• Kernel reveals vectors eliminated by transformation
• Range shows possible outputs of transformation
• Rank indicates dimensionality of transformed space
• Nullity measures information loss during transformation
• Rank-nullity theorem connects domain, range, and kernel dimensions
• Linear transformations form foundation for analyzing vector space mappings

Examples of linear transformations

Geometric transformations

• Rotation in two-dimensional space preserves angles and distances between points (90-degree rotation)
• Scaling multiplies all coordinates by constant factor, changing size but not shape (doubling all dimensions)
• Reflection across line or plane flips objects over specified axis or surface (mirror image)
• Shear transformations shift points parallel to given line or plane proportionally to perpendicular distance (slanting a rectangle into a parallelogram)
• Projection onto subspace maps vectors to lower-dimensional space (shadow cast by 3D object onto 2D surface)

Applied linear transformations

• Computer graphics use linear transformations to manipulate objects in 2D and 3D space (rotating a 3D model)
• Signal processing applies convolution with fixed function for filtering (smoothing audio signal)
• Image processing utilizes linear transformations for various effects (blurring or sharpening an image)
• Economics models employ linear transformations to analyze input-output relationships (Leontief model)
• Quantum mechanics describes particle state changes using linear operators (time evolution of wave function)

Applying linear transformations

Matrix multiplication method

• Apply linear transformation to vector by multiplying transformation matrix with vector as column matrix
• Resulting vector represents image of input vector in codomain space
• Matrix-vector multiplication combines linear combination of matrix columns
• Order of multiplication matters: matrix must be on left, vector on right
• Dimensions must match: number of matrix columns equals vector length

Geometric interpretation

• Linear transformations can change direction, magnitude, or both of input vector
• Transformation effect on unit circle or sphere reveals impact on all vectors
• Stretching, compressing, rotating, or reflecting unit circle indicates transformation properties
• Determinant of transformation matrix indicates preservation, reversal, or collapse of orientation and volume
• Positive determinant preserves orientation, negative reverses, zero collapses space

Advanced concepts

• Linear transformations can change basis of vector space
  • Useful for simplifying calculations (diagonalization)
  • Reveals underlying structure (principal component analysis)
• Eigenvectors of linear transformation only scale, not rotate
  • Corresponding eigenvalues indicate scaling factor
  • Eigenvectors and eigenvalues provide insight into transformation behavior
• Singular value decomposition decomposes transformation into rotation, scaling, and another rotation
  • Reveals principal directions and magnitudes of transformation

Composition of linear transformations

Properties of composition

• Composition of two linear transformations produces another linear transformation
• Matrix multiplication represents composition of transformations represented by matrices
• Multiply matrices in order of application (right to left) to compose transformations
• Resulting matrix represents new linear transformation combining effects of individual transformations
• Order of composition matters for most linear transformations (matrix multiplication not generally commutative)
• Associativity of matrix multiplication ensures (AB)C = A(BC) for transformation matrices A, B, C

Inverse transformations and decomposition

• Inverse of composition equals composition of inverses in reverse order: (T₁ ∘ T₂)⁻¹ = T₂⁻¹ ∘ T₁⁻¹
• Composing transformation with its inverse yields identity transformation
• Identity transformation leaves vectors unchanged (represented by identity matrix)
• Decomposition breaks complex transformations into simpler, more manageable steps
  • Rotation can be decomposed into shears (shear-rotate-shear decomposition)
  • 3D rotations can be expressed as compositions of rotations around coordinate axes

Applications of composition

• Computer graphics use composition to create complex animations (sequence of rotations, scales, and translations)
• Robot kinematics apply compositions of transformations to describe joint movements
• Coordinate transformations in physics often involve composing multiple reference frame changes
• Signal processing chains compose multiple filtering operations as series of linear transformations
• Machine learning algorithms may compose multiple linear layers in neural networks