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Mathematics LibreTexts

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 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 v1,v2,,vn is linearly independent, we must show that the equation c1v1+c2v2++cnvn=0 has no solutions other than c1=c2==cn=0.

Example 10.2.1:

Consider the following vectors in 3:
v1=(002),v2=(221),v3=(143).

Are they linearly independent?

We need to see whether the system

c1v1+c2v2+c3v3=0

has any solutions for c1,c2,c3. We can rewrite this as a homogeneous system:

(v1v2v3)(c1c2c3)=0.

This system has solutions if and only if the matrix M=(v1v2v3) is singular, so we should find the determinant of M:

detM=det(021024213)=2det(2124)=12.

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

(v1v2v3)(c1c2c3)=0

is c1=c2=c3=0. So the vectors v1,v2,v3 are linearly independent.

Contributor

This page titled 10.2: Showing Linear Independence is shared under a not declared license and was authored, remixed, and/or curated by David Cherney, Tom Denton, & Andrew Waldron.

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