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Mathematics-Online course: Linear Algebra - Matrices - Determinants

Basis Test

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The columns of a quadratic matrix $ A$ are linearly independent if and only if $ \operatorname{det}A\ne 0$.

(Authors: Burkhardt/Höllig/Hörner)
If the first row of $ A$ consists only of zeros, then the statement is obvious. Otherwise, we can assume that $ a_{1,1}\ne 0$ (since we can interchange the columns). Now, modify the columns $ a_j$ of $ A$ as follows:

$\displaystyle a_j \to a'_j = a_j - \frac{a_{1,j}}{a_{1,1}}\, a_1, \quad j=2,\ldots,n\, . $

These operations do not change the determinant and if the columns of $ A$ form a basis then so do the modified columns. The modified matrix has the form

$\displaystyle A' = \left(\begin{array}{cccc} a_{1,1} & 0 & \cdots & 0 \\ c & & B \end{array}\right)\, . $

By the expansion formula we have

$\displaystyle \operatorname{det}A' = a_{1,1} \operatorname{det}B\, , $

and, since $ a_{1,1}\ne 0$, the columns of $ A'$ form a basis of $ K^n$ if and only if the columns of $ B$ form a basis of $ K^{n-1}$. Hence, the statement can be proved by induction.

(Authors: Burkhardt/Höllig/Hörner)
By the example of matrix

$\displaystyle A=\left(\begin{array}{cc} a&b \\ c&d \end{array}\right) $

the basis test can be clarified.

If one row or column consists only of zeros, then the determinant equals zero. In this case the rank of the matrix is at most 1 and, hence, the columns and the rows, resp., of the matrix can not form a basis.

On the other hand, from $ ad=bc$ it follows that

$\displaystyle a\left(\begin{array}{r}b\\ d\end{array}\right)-b \left(\begin{ar... ...c\end{array}\right) =\left(\begin{array}{r}ab-ba\\ ad-bc\end{array}\right)=0 $

and, hence, the columns are linearly dependent.

Thus, if the columns of $ A$ are linearly dependent, we have

$\displaystyle \left(\begin{array}{r}b\\ d\end{array}\right)= \lambda\left(\begin{array}{r}a\\ c\end{array}\right)\,, $

and therefore the calculation of the determinant yields

$\displaystyle \operatorname{det}A=ad-bc=a\lambda c - \lambda a c =0\,. $

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005