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Subspaces associated to a matrix

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Definition of matrix null space (right)
Definition of matrix null space (left)
Definition of column space of a matrix
The column space of an m-by-n matrix is a subspace of R^m (or C^m)
The pivot columns of a matrix are a basis for the column space.
Row operations do not necessarily preserve the column space.
Nonzero rows of reduced row-echelon form give a basis for the row space
Definition of row space of a matrix
Row equivalent matrices have the same row space.

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The column space of an m-by-n matrix is a subspace of R^m (or C^m)

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Definition of column space of a matrixClose match
The column space of a matrix is a vector spaceClose match
Subspaces associated to a matrixClose match
The left null space of a matrix is a subspace of R^m (or C^m).Close match
The row space of a matrix is a vector spaceClose match
The null space of a matrix is a subspace of R^n (or C^n).Close match
Definition of row space of a matrixClose match
The set of m by n matrices is a vector space.Close match
Definition of matrix null space (left)Close match
Row operations do not necessarily preserve the column space.Close match
Definition of matrix multiplication in terms of column vectorsClose match
Definition of matrix null space (right)Close match
Null spaces are orthogonal complements of the corresponding row or column spacesClose match
The pivot columns of a matrix are a basis for the column space.Close match
Row equivalent matrices have the same row space.Close match
Nonzero rows of reduced row-echelon form give a basis for the row spaceClose match
An eigenspace of a matrix is the null space of a related matrix.Close match
An eigenspace of a matrix is a nontrivial subspace.Related
Definition of eigenspace of a matrixRelated
An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).Related

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