10.2: Showing Linear Independence
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 ℜ3:
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
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)