Mathematics-Online course: Linear Algebra-Matrices-Matrix Operations-Rank of a Matrix

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	Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations
	Rank of a Matrix

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For an $m\times n$ -matrix

the maximal number of linearly independent columns equals the maximal number of linearly independent rows. This number is called rank of

and is denoted by $\operatorname{Rg}A$ .

The maximal number of linearly independent colums $v_1,\ldots,v_n$ corresponds to the dimension of the vector space

$\displaystyle V = \operatorname{span}\{v_1,\ldots,v_n\}\, ,$

since a basis can be selected from the columns. An analogous statement holds true for the vector space

spanned by the rows $w_1,\ldots,w_m$ . The assertion

$\displaystyle \operatorname{dim}V = \operatorname{dim}W$

will now be proved by induction on

It can easily be seen that $\operatorname{dim}V$ and $\operatorname{dim}W$ remain unchanged under the following operations:

Permutation of columns or rows;
Addition of a scalar multiple of a column (row) to another column (row).

If matrix $A\ne 0$ then, by the operations given above, this matrix can be brought to the form

$\displaystyle B = \left(\begin{array}{cccc} b_{1,1} & 0 & \ldots & 0 \\ 0 \\ \vdots & & A' \\ 0 \end{array}\right)\, .$

At first we achieve $b_{1,1}\ne0$ by permutations. Then the remaining elements of the first row and column are nullified by appropriate additions. More precisely, we add the $(-b_{1,j}/b_{1,1})$ -fold of the first column to the

-th column ( $j=2,\ldots,n$ ) and then we proceed with the rows in an analogous way. Let

and

denote the respective vector spaces for the $(m-1)\times(n-1)$ -matrix

, then we obtain

$\displaystyle \operatorname{dim} V = \operatorname{dim} V' + 1, \quad \operatorname{dim} W = \operatorname{dim} W' + 1$

and the assertion follows by induction.

(Authors: Burkhardt/Höllig/Hörner)

Some typical cases are illustrated by means of three $3\times2$ -matrices:

$\displaystyle \operatorname{Rg}\left(\begin{array}{rr} 0 & 0 \\ 0 & 0 \\ 0... ...\begin{array}{rr} 3 & 4 \\ 6 & 8 \\ 9 & 12 \end{array} \right) =1 \,.$

For the third matrix we have for the columns and , and for the rows and . Thus in either case the dimension of the corresponding vector space equals 1.

(Authors: Burkhardt/Höllig/Hörner)

automatically generated 4/21/2005