Linear Transformation in Algebra

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: 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

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

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

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

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

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

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

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

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

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

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

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