Rank of a matrix using determinants (1).pdf

# Rank of a matrix using determinants (1).pdf - Rank of a...

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Rank of a matrix using determinants We begin with some definitions: The row space of a matrix is the subspace generated by the row vectors and the column space of a matrix is the subspace generated by the column vectors. It can be proved that the dimension of both these spaces is the same and is called the rank of the matrix . Thus rank is the maximum number of linearly independent columns(rows) of a matrix. We can find the rank from the row echelon form of the matrix and we can also find the rank using determinants. For this it is enough to know that n vectors in R n are linearly independent if and only if the determinant of the matrix obtained by considering these n vectors as columns is non-zero . This is precisely the theorem 3.1 on page 210 of the text book with a small typo!!!! The proof of the theorem is not very difficult if you understand that the theorem can be restated as n vectors in R n are linearly dependent if and only if the determinant of the matrix obtained by considering these n
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