Mathematics Foundations/6.6 Linear Transformations

6.6 Linear Transformations Introduction Linear transformations are fundamental mathematical objects that preserve the operations of vector addition and scalar multiplication. They provide a powerful framework for understanding how vectors are mapped from one vector space to another while maintaining linearity properties. Definition and Basic Properties Let and be vector spaces over the same field (typically or ). A function is called a linear transformation if for all vectors and all scalars , the following two conditions are satisfied: Additivity: Homogeneity: These two conditions can be combined into a single condition: for all and . Examples of Linear Transformations Identity transformation: for all Zero transformation: for all Rotation in : rotates vector by angle Projection onto a subspace Reflection across a plane or line Scaling: for a fixed scalar Differentiation: is a linear transformation on function spaces Integration: is a linear transformation from function spaces to Matrix Representation of Linear Transformations When working with finite-dimensional vector spaces with specified bases, every linear transformation can be represented by a matrix. Let be a linear transformation, where has basis and has basis . The matrix representation of with respect to these bases is the matrix whose columns are the coordinate vectors of expressed in the basis of . Standard Matrix of a Linear Transformation If and with standard bases, then the standard matrix of is defined by: where is the th standard basis vector. For any vector , we have: Change of Basis If is the change-of-basis matrix from basis to the standard basis in , and is the change-of-basis matrix from the standard basis to basis in , then: where is the matrix of with respect to bases and , and is the standard matrix of . Kernel and Image Kernel (Null Space) The kernel (or null space) of a linear transformation is the set of all vectors in that maps to the zero vector in : Properties of the kernel: is a subspace of is injective (one-to-one) if and only if If is represented by matrix , then Image (Range) The image (or range) of a linear transformation is the set of all possible outputs: Properties of the image: is a subspace of is surjective (onto) if and only if If is represented by matrix , then The Rank-Nullity Theorem For a linear transformation where is finite-dimensional: This fundamental result connects the dimension of the domain with the dimensions of the kernel and image. In matrix terms, if is the matrix of : Types of Linear Transformations Isomorphism A linear transformation is an isomorphism if it is both injective and surjective. In this case: and have the same dimension There exists a linear transformation such that and If is represented by a square matrix , then is invertible Endomorphism A linear transformation from a vector space to itself is called an endomorphism. Automorphism An automorphism is an isomorphism from a vector space to itself. Projection A linear transformation is a projection if (i.e., applying twice is the same as applying it once). Eigenvalues and Eigenvectors of Linear Transformations For an endomorphism , a non-zero vector is an eigenvector of with eigenvalue if: The set of all eigenvectors corresponding to an eigenvalue , together with the zero vector, forms a subspace called the eigenspace of . Diagonalization An endomorphism is diagonalizable if there exists a basis of consisting entirely of eigenvectors of . In matrix terms, if is the matrix of with respect to the standard basis, then is diagonalizable if there exists an invertible matrix such that is diagonal. Linear Transformations in Geometry Rotations In , a rotation by angle counterclockwise about the origin is represented by the matrix: Reflections A reflection across a line through the origin with unit direction vector is represented by: Shears A shear transformation in parallel to the x-axis is represented by: Projections The orthogonal projection onto a subspace with orthonormal basis is: Applications of Linear Transformations Computer Graphics Linear transformations are fundamental in computer graphics for operations such as: Scaling, rotation, and reflection of objects Perspective projections Coordinate transformations Signal Processing Linear transformations are used in: Fourier transforms to convert signals between time and frequency domains Wavelet transforms for multi-resolution analysis Digital filters Quantum Mechanics In quantum mechanics, observables are represented by linear operators (transformations) acting on state vectors in Hilbert space. Machine Learning Linear transformations appear in: Principal Component Analysis (PCA) Linear regression models Neural network layers Exercises Prove that the composition of two linear transformations is also a linear transformation. Find the standard matrix of the linear transformation that rotates vectors by counterclockwise and then reflects them across the y-axis. Determine the kernel and image of the linear transformation represented by the matrix . Let be the linear transformation that maps a polynomial to its derivative. Find the kernel and image of . Let be the linear transformation defined by . Find a basis for the kernel and image of . Category:Linear Algebra Category:Linear Transformations Category:Vector Spaces