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Rank and nullity

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Definition of column rank of a matrix
Definition of rank of a matrix
Definition of nullity of a matrix
The rank of a matrix equals the rank of the linear transformation it represents.
The row space and the column space of a matrix have the same dimension.
The rank of a matrix equals number of pivots in a reduced row echelon form.
Rank–nullity theorem for an m-by-n matrix

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The row space and the column space of a matrix have the same dimension.

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The row space of a matrix is a vector spaceClose match
The rank of a matrix equals number of pivots in a reduced row echelon form.Close match
Row equivalent matrices have the same row space.Close match
The column space of a matrix is a vector spaceClose match
The rank of a matrix equals the rank of the linear transformation it represents.Close match
Definition of rank of a matrixClose match
Every matrix is row-equivalent to only one matrix in reduced row echelon form.Close match
Definition of column rank of a matrixClose match
Two matrices of the same size are equivalent if and only if they have the same rank.Close match
Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrixClose match
Definition of row space of a matrixClose match
Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.Close match
Rank and nullityClose match
Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.Close match
A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.Close match
Rank–nullity theorem for an m-by-n matrixClose match
Equivalence theorem for nonsingular matrices: the matrix A has rank n.Close match
Every matrix is row-equivalent to a matrix in reduced row echelon form.Close match
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The null space of a matrix is a subspace of R^n (or C^n).Close match

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