The dimension of a finite-dimensional VectorSpace, V, is the number of vectors in the VectorSpaceBasis for V. The zero VectorSpace has dimension zero. R^n has an n-dimensional VectorSpace. Reducing a Matrix to RowEchelon form (by GaussianElimination) allows a basis for the RowSpace to be extracted from the non-zero rows. If this Basis has n rows, the RowSpace is n-dimensional.
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See also MatrixRank

CategoryMath