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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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Nonzero rows of reduced row-echelon form give a basis for the row space

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Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrixClose match
Definition of reduced row echelon form of a matrixClose match
Every matrix is row-equivalent to only one matrix in reduced row echelon form.Close match
Description of a spanning set for the null space of a matrix from the reduced row-echelon form.Close match
Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrixClose match
Echelon matricesClose match
The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.Close match
Definition of matrix in reduced row echelon formClose match
Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrixClose match
Example of row reducing a 3-by-3 matrixClose match
Every matrix is row-equivalent to a matrix in reduced row echelon form.Close match
Definition of (row) echelon form of a matrixClose match
Description of a basis for the null space of a matrix from the reduced row-echelon form.Close match
Row equivalent matrices have the same row space.Close match
The nonzero rows of an echelon form of a matrix are linearly independent.Close match
The row space of a matrix is a vector spaceClose match
Row operations do not necessarily preserve the column space.Close match
Definition of row space of a matrixClose match
Subspaces associated to a matrixClose match
The pivot columns of a matrix are a basis for the column space.Close match

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