Linear Transformation in Algebra

Linear Transformation

Matrix Multiplication

• A linear transformation can be represented as a matrix multiplication
• Matrix multiplication is not commutative, i.e., AB ≠ BA
• The number of columns in the first matrix must match the number of rows in the second matrix
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix

Image and Kernel

• Image: The set of all output vectors resulting from the linear transformation
• Kernel (or Null Space): The set of all input vectors that result in the zero output vector
• The kernel is a subspace of the domain, and the image is a subspace of the codomain
• The kernel and image are orthogonal to each other

Vector Operations

• Scaling: A linear transformation can scale a vector by a scalar value
• Reflection: A linear transformation can reflect a vector across a line or plane
• Projection: A linear transformation can project a vector onto a line or plane
• Rotation: A linear transformation can rotate a vector by a certain angle

Determinants

• The determinant of a matrix represents the scaling factor of the linear transformation
• A determinant of 0 indicates that the linear transformation is not invertible (i.e., it's not one-to-one)
• A determinant of 1 indicates that the linear transformation preserves the magnitude of the input vectors
• The determinant can be used to find the inverse of a matrix, if it exists

Linear Transformation

Matrix Representation

• A linear transformation can be represented as a matrix multiplication, which is a powerful tool for performing transformations
• However, matrix multiplication is not commutative, meaning the order of matrices matters: AB ≠ BA
• For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix

Image and Kernel

• The image of a linear transformation is the set of all possible output vectors
• The kernel (or null space) of a linear transformation is the set of all input vectors that result in the zero output vector
• Both the kernel and image are subspaces, with the kernel being a subspace of the domain and the image being a subspace of the codomain
• The kernel and image are orthogonal to each other, meaning they have a 90-degree angle between them

Effects on Vectors

• A linear transformation can scale a vector by a scalar value, changing its magnitude
• A linear transformation can reflect a vector across a line or plane, changing its direction
• A linear transformation can project a vector onto a line or plane, changing its direction and magnitude
• A linear transformation can rotate a vector by a certain angle, changing its direction

Determinants

• The determinant of a matrix represents the scaling factor of the linear transformation it represents
• A determinant of 0 indicates that the linear transformation is not invertible (not one-to-one)
• A determinant of 1 indicates that the linear transformation preserves the magnitude of the input vectors
• The determinant can be used to find the inverse of a matrix, if it exists, allowing us to reverse the transformation

Linear Independence

• A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 +...+ cnvn = 0 is c1 = c2 =...= cn = 0
• Linear independence means that a linear combination of vectors results in the zero vector only if all coefficients are zero

Linear Transformations

• A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication
• A linear transformation can be represented by a matrix, and its matrix representation is unique for a given basis
• The kernel is the set of vectors that map to the zero vector, while the image is the set of vectors that can be obtained by applying the transformation
• The rank is the dimension of the image, and the nullity is the dimension of the kernel

Span and Basis

• The span of a set of vectors is the set of all linear combinations of the vectors
• A basis is a set of vectors that spans the vector space and is linearly independent
• A basis can be used to represent every vector in the vector space, and the dimension of a vector space is the number of vectors in a basis
• A standard basis consists of unit vectors aligned with the coordinate axes

Dimension and Rank

• The dimension of a vector space is the number of vectors in a basis
• The rank of a matrix is the maximum number of linearly independent rows or columns
• The nullity of a matrix is the number of linearly independent solutions to the equation Ax = 0
• The rank-nullity theorem states that the rank of a matrix plus the nullity of a matrix is equal to the number of columns
• The dimension theorem states that the dimension of a vector space is equal to the rank of a matrix representation of a linear transformation

Eigenvalues and Eigenvectors

• An eigenvalue is a scalar that satisfies the equation Ax = λx for some non-zero vector x
• An eigenvector is a non-zero vector that satisfies the equation Ax = λx for some scalar λ
• The eigenvalue equation is Ax = λx, and the characteristic equation is det(A - λI) = 0
• The eigendecomposition of a matrix is a diagonal matrix consisting of the eigenvalues and a matrix of eigenvectors