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Matrix/Column rank/Row rank/Introduction/Section

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Let be a field, and let denote an -matrix over . Then the dimension of the linear subspace of , generated by the columns, is called the column rank of the matrix, written


Let denote a field, and let and denote -vector spaces of dimensions and . Let

be a linear mapping which is described by the matrix , with respect to bases of the spaces. Then

holds.

Proof


To formulate the next statement, we introduce row rank of an -matrix to be the dimension of the linear subspace of generated by the rows.


Let be a field, and let denote an -matrix over . Then the column rank coincides with the row rank. The rank equals the number from

fact.

In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. The row rank of equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank , since the first rows are linearly independent, and, apart from this, this, there are only zero rows. It has also column rank , since the columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these columns. By exercise, the column rank is preserved by elementary row manipulations.


Both ranks coincide, so we only talk about the rank of a matrix.


Let be a field, and let denote an -matrix

over . Then the following statements are equivalent.
  1. is invertible.
  2. The rank of is .
  3. The rows of are linearly independent.
  4. The columns of are linearly independent.
This follows from fact and from fact.