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- 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 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 matrix representation of the inverse of linear transformations is the inverse of the matricix.#24074
- 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 nonzero rows of the reduced row-echelon form of a matris are a basis for the row space.#23879
- The null space of a matrix is a subspace of R^n (or C^n).#24110
- The null space of a matrix is the orthogonal complement of the column space.#23924
- 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
- Theorem describing matrix multiplication#23490
- Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix#23551
- Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23552
- Theorem describing the determinants of elementary matrices.#23745
- Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23553
- Theorem describing the vector form of sulutions to a linear system.#23444
- The orthogonal complement of a subspace is a subspace.#23921
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