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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 pivot columns of a matrix are a basis for the column space.

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Definition of pivot columnClose match
Row operations do not necessarily preserve the column space.Close match
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.Close match
Example of putting a matrix in echelon form and identifying the pivot columnsClose match
Definition of pivot positionClose match
Definition of pivotClose match
A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.Close match
Nonzero rows of reduced row-echelon form give a basis for the row spaceClose match
Row equivalent matrices have the same row space.Close match
Example of (echelon matrix/matrix in (row) echelon form)Close match
Definition of column space of a matrixClose match
The column space of a matrix is a vector spaceClose match
The column space of an m-by-n matrix is a subspace of R^m (or C^m)Close match
Subspaces associated to a matrixClose match
Description of a basis for the null space of a matrix from the reduced row-echelon form.Close match
Definition of row space of a matrixRelated
An eigenspace of a matrix is the null space of a related matrix.Related
An eigenspace of a matrix is a nontrivial subspace.Related
A linear transformation is given by a matrix with respect to a given basis.Related
Definition of eigenspace of a matrixRelated

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