Linear transformations
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- Eigenvalues and eigenvectors
- Terminology
- Geometric properties of linear transformations
- Matrices as linear transformations
- Basic properties of linear transformations
- Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
- Description of a basis for the null space of a matrix from the reduced row-echelon form.
- The nonzero rows of an echelon form of a matrix are linearly independent.
- Subspaces associated to a matrix
- Rank and nullity
- Examples
- Composition
- The preimage of a vector is a translation of the kernel of the linear transformation
- The image of a linearly independent set under an injective linear transformation is linearly independent.
- The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
- The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
- A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
- The range of a linear transformation is a subspace
- The the image of a spanning set is a spanning set for the range space
- A linear transformation is surjective if and only if the image of a basis is a spanning set
- Definition of generalized range space of a linear transformation
- A linear transformation is injective on its generalized range space.
- Definition of diagonalizable linear transformation
- A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
- Subspaces associated to a linear transformation
- The rank plus the nullity of a linear transformation equals the dimension of the domain.
- The image of a linearly dependent set under a linear transformation is linearly dependent.
- A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
- A linear transformation is invertible if and only if it is injective and surjective
- Definition of matrix representation of a linear transformation with respect to bases of the spaces
- A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
- Definition of matrix representation of a linear transformation from a vector space to itself
- The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
- The matrix representation of a sum of linear transformations is the sum of the matrices.
- The matrix representation of a composition of linear transformations is the product of the matrices.
- The matrix representation of the inverse of linear transformations is the inverse of the matricix.
- A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
- A linear transformation has a representation as an upper triangular matrix.
- Equivalence theorems for injective transformations
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