Abstract Linear Algebra II Unit 2 Review: Linear Transformations

Key Concepts and Definitions

• Linear transformations map vectors from one vector space to another while preserving linear combinations and the zero vector
• Domain refers to the vector space where the linear transformation starts, and codomain is the vector space where it ends up
• Isomorphisms are bijective linear transformations with linearly independent columns in their matrix representation
• Endomorphisms are linear transformations from a vector space to itself
• Automorphisms are invertible endomorphisms, meaning they have an inverse transformation that "undoes" the original
• Eigenvalues are scalars $\lambda$ that satisfy the equation $T(v) = \lambda v$ for some nonzero vector $v$ and linear transformation $T$
  • The corresponding nonzero vectors $v$ are called eigenvectors
• Diagonalization expresses a linear transformation as a diagonal matrix, which simplifies computations and analysis

Vector Spaces and Linear Maps

• Vector spaces are sets that are closed under vector addition and scalar multiplication, with a zero vector
• Linear maps, or linear transformations, preserve the vector space structure when mapping between two vector spaces
  • They satisfy $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$ for vectors $u, v$ and scalar $c$
• The kernel of a linear transformation $T$ is the set of vectors that map to the zero vector, $\ker(T) = \{v : T(v) = 0\}$
• The image or range of $T$ is the set of all vectors in the codomain that $T$ maps to, $\text{Im}(T) = \{T(v) : v \text{is in the domain}\}$
• Rank-nullity theorem states that for a linear map $T: V \to W$, $\dim(V) = \dim(\ker(T)) + \dim(\text{Im}(T))$
• Isomorphic vector spaces have the same dimension and are structurally identical, even if they appear different

Properties of Linear Transformations

• Linearity is the defining property, meaning preserving vector addition and scalar multiplication
• Injective (one-to-one) linear transformations have a trivial kernel, containing only the zero vector
• Surjective (onto) linear transformations have an image equal to the entire codomain
• Bijective linear transformations are both injective and surjective, and have an inverse transformation
• Compositions of linear transformations are linear, i.e., if $S$ and $T$ are linear, then $S \circ T$ is linear
• The identity transformation $I(v) = v$ maps each vector to itself and is linear
• The zero transformation maps every vector to the zero vector and is linear

Matrix Representations

• Every linear transformation can be represented by a matrix with respect to chosen bases for the domain and codomain
• The matrix $A$ of a linear transformation $T$ satisfies $T(v) = Av$ for all vectors $v$ in the domain
• Changing bases corresponds to similarity transformations on the matrix, $A' = P^{-1}AP$, where $P$ is the change of basis matrix
• Matrix multiplication represents the composition of linear transformations
• The determinant of the matrix determines if the transformation is invertible (nonzero determinant) or not (zero determinant)
• Eigenvalues and eigenvectors can be found using the characteristic equation $\det(A - \lambda I) = 0$
• Diagonalization is possible when there is a basis of eigenvectors, resulting in a diagonal matrix representation

Kernel and Image

• The kernel is the set of all vectors that map to the zero vector under the transformation
  • It forms a subspace of the domain and its dimension is called the nullity
• The image is the set of all vectors in the codomain that are outputs of the transformation
  • It forms a subspace of the codomain and its dimension is called the rank
• Rank-nullity theorem relates the dimensions of the kernel, image, and domain: $\dim(V) = \dim(\ker(T)) + \dim(\text{Im}(T))$
• Injectivity is equivalent to having a trivial kernel (only contains the zero vector)
• Surjectivity is equivalent to the image being equal to the entire codomain
• The first isomorphism theorem states that a linear transformation $T: V \to W$ induces an isomorphism between $V/\ker(T)$ and $\text{Im}(T)$

Eigenvalues and Eigenvectors

• Eigenvectors are nonzero vectors that, when transformed, result in a scalar multiple of themselves
  • The scalar multiple is called the eigenvalue, satisfying $T(v) = \lambda v$
• Eigenspaces are the sets of all eigenvectors corresponding to a specific eigenvalue, together with the zero vector
• The characteristic equation $\det(A - \lambda I) = 0$ is used to find eigenvalues, where $A$ is the matrix of the transformation
• Algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic equation
• Geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace
• Diagonalizable matrices have a basis of eigenvectors and can be written as $A = PDP^{-1}$, where $D$ is diagonal and $P$ contains eigenvectors

Applications and Examples

• Markov chains use stochastic matrices to model transitions between states, with eigenvalues and eigenvectors providing long-term behavior
• Quantum mechanics represents states as vectors and observables as linear operators, with eigenvalues and eigenvectors corresponding to measurable quantities
• Computer graphics use linear transformations for scaling, rotation, reflection, and shearing of images
  • Homogeneous coordinates allow for the representation of translations as matrix operations
• Fourier transforms decompose functions into sums of simpler trigonometric functions, which is a change of basis
• Least squares fitting finds the best linear approximation to data by minimizing the sum of squared errors
• Differential equations can be solved using eigenvalues and eigenvectors of the associated linear operator
• Principal component analysis (PCA) uses eigenvectors of the covariance matrix to find the most informative directions in data

Common Pitfalls and Tips

• Ensure that transformations are well-defined and linear by checking the properties on the entire domain
• Be careful when using matrix representations, as they depend on the choice of bases for the domain and codomain
• Eigenvalues and eigenvectors are only defined for square matrices, so the domain and codomain must have the same dimension
• Not all matrices are diagonalizable; they must have a full set of linearly independent eigenvectors
• The zero vector is not an eigenvector, even though it satisfies the eigenvector equation for any eigenvalue
• Algebraic and geometric multiplicities of eigenvalues can differ, with geometric multiplicity always less than or equal to algebraic multiplicity
• When using linear transformations in applications, be aware of any assumptions or limitations of the model
• Practice visualizing linear transformations in 2D and 3D to build intuition, then generalize to higher dimensions