Editing Linear independence (section)

=== Alternative method using determinants ===

An alternative method relies on the fact that <math>n</math> vectors in <math>\mathbb{R}^n</math> are linearly '''independent''' [[if and only if]] the [[determinant]] of the [[matrix (mathematics)|matrix]] formed by taking the vectors as its columns is non-zero.

In this case, the matrix formed by the vectors is
:<math>A = \begin{bmatrix}1&-3\\1&2\end{bmatrix} .</math>
We may write a linear combination of the columns as
:<math>A \Lambda = \begin{bmatrix}1&-3\\1&2\end{bmatrix} \begin{bmatrix}\lambda_1 \\ \lambda_2 \end{bmatrix} .</math>
We are interested in whether {{math|1=''A''Λ = '''0'''}} for some nonzero vector Λ. This depends on the determinant of <math>A</math>, which is
:<math>\det A = 1\cdot2 - 1\cdot(-3) = 5 \ne 0.</math>
Since the [[determinant]] is non-zero, the vectors <math>(1, 1)</math> and <math>(-3, 2)</math> are linearly independent.

Otherwise, suppose we have <math>m</math> vectors of <math>n</math> coordinates, with <math>m < n.</math> Then ''A'' is an ''n''×''m'' matrix and Λ is a column vector with <math>m</math> entries, and we are again interested in ''A''Λ&nbsp;= '''0'''. As we saw previously, this is equivalent to a list of <math>n</math> equations. Consider the first <math>m</math> rows of <math>A</math>, the first <math>m</math> equations; any solution of the full list of equations must also be true of the reduced list. In fact, if {{math|⟨''i''<sub>1</sub>,...,''i''<sub>''m''</sub>⟩}} is any list of <math>m</math> rows, then the equation must be true for those rows.
:<math>A_{\lang i_1,\dots,i_m \rang} \Lambda = \mathbf{0} .</math>
Furthermore, the reverse is true. That is, we can test whether the <math>m</math> vectors are linearly dependent by testing whether
:<math>\det A_{\lang i_1,\dots,i_m \rang} = 0</math>
for all possible lists of <math>m</math> rows. (In case <math>m = n</math>, this requires only one determinant, as above. If <math>m > n</math>, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.