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# 10.2: Showing Linear Independence

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We have seen two different ways to show a set of vectors is linearly dependent: we can either find a linear combination of the vectors which is equal to zero, or we can express one of the vectors as a linear combination of the other vectors. On the other hand, to check that a set of vectors is linearly $$\textit{independent}$$, we must check that every linear combination of our vectors with non-vanishing coefficients gives something other than the zero vector. Equivalently, to show that the set $$v_{1}, v_{2}, \ldots, v_{n}$$ is linearly independent, we must show that the equation $$c_{1} v_{1}+c_{2} v_{2} + \cdots + c_{n} v_{n}=0$$ has no solutions other than $$c_{1}=c_{2}=\cdots=c_{n}=0.$$

Example 109

Consider the following vectors in $$\Re^{3}$$:
$v_{1}=\begin{pmatrix}0\\0\\2\end{pmatrix}, \qquad v_{2}=\begin{pmatrix}2\\2\\1\end{pmatrix}, \qquad v_{3}=\begin{pmatrix}1\\4\\3\end{pmatrix}.$

Are they linearly independent?

We need to see whether the system

$c^{1}v_{1} + c^{2}v_{2}+ c^{3}v_{3}=0$

has any solutions for $$c^{1}, c^{2}, c^{3}$$. We can rewrite this as a homogeneous system:

$\begin{pmatrix}v_{1}&v_{2}&v_{3}\end{pmatrix}\begin{pmatrix}c^{1}\\c^{2}\\c^{3}\end{pmatrix}=0.$

This system has solutions if and only if the matrix $$M=\begin{pmatrix}v_{1}&v_{2}&v_{3}\end{pmatrix}$$ is singular, so we should find the determinant of $$M$$:

$\det M = \det \begin{pmatrix} 0 & 2 & 1 \\ 0 & 2 & 4 \\ 2 & 1 & 3 \\ \end{pmatrix} = 2 \det \begin{pmatrix} 2 & 1 \\ 2 & 4 \\ \end{pmatrix} =12.$

Since the matrix $$M$$ has non-zero determinant, the only solution to the system of equations

$\begin{pmatrix}v_{1}&v_{2}&v_{3}\end{pmatrix}\begin{pmatrix}c^{1}\\c^{2}\\c^{3}\end{pmatrix}=0$

is $$c_{1}=c_{2}=c_{3}=0$$. So the vectors $$v_{1}, v_{2}, v_{3}$$ are linearly independent.