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- The inverse of a matrix can be used to solve a linear system.#23532
- The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.#23527
- The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.#23565
- The inverse of an isomorphism is an isomorphism.#24002
- The inverse of a scalar multiple is the reciprocal times the inverse.#23525
- The inverse of the inverse of a linear transformation is the original linear transformation#24029
- The kernel/null space of a linear transformation is a subspace#24059
- The kernels of powers of a linear transformation form an ascending chain#24062
- The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.#23950
- The left null space of a matrix is a subspace of R^m (or C^m).#24111
- The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.#23768
- The matrix of an inverse is the inverse of the matrix representation#24074
- The matrix representation of a composition of linear transformations is the product of the matrices.#24073
- The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.#24071
- The matrix representation of a sum of linear transformations is the sum of the matrices.#24072
- The minimal polynomial of a linear transformation exists and is unique.#24047
- The minimal polynomial of a square matrix exists and is unique.#23704
- The nonzero rows of an echelon form of a matrix are linearly independent.#23871
- The null space of a matrix is a subspace of R^n (or C^n).#24110
- The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.#23774
- The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.#23773
- The number of solutions to a linear system#23438
- Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors#23820
- Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0#23822
- Theorem characterizing when a space is the direct sum of two subspaces#23976