Vector spaces · Definition

Linear independence

linear algebravector spaces

Before you start you need to know

It detects when vectors get in each other's way: if one is redundant, the set is dependent.

Definition

A set of vectors is linearly independent if the only linear combination of them that gives the zero vector is the one with all coefficients equal to zero.

\[c_1v_1+\cdots+c_kv_k=0\ \Rightarrow\ c_1=\cdots=c_k=0\]

The combination with all coefficients zero always gives \(0\); the question is whether another one does. If so, the set is linearly dependent. For example, \(\{(1,2),(2,4)\}\) is dependent because \(2(1,2)-(2,4)=0\), while \(\{(1,0),(0,1)\}\) is independent: \(c_1(1,0)+c_2(0,1)=(c_1,c_2)=0\) forces \(c_1=c_2=0\).

In \(\mathbb{R}^n\) you decide this by setting up the homogeneous system \(c_1v_1+\cdots+c_kv_k=0\) and applying Gaussian elimination: the set is independent exactly when the only solution is the trivial one. Consequently, more than \(n\) vectors in \(\mathbb{R}^n\) are always dependent, since there are more unknowns than equations.

Dependent means that at least one vector is a linear combination of the others: it can be solved for from a nontrivial relation and removed without changing the span. A set containing the zero vector is always dependent, and two vectors are dependent exactly when one is a multiple of the other.

Typical mistake: believing it is enough to check pairs of vectors. The vectors \((1,0,1)\), \((0,1,1)\) and \((1,1,2)\) have no two proportional, yet they are dependent because the third is the sum of the first two. Independence, together with span, defines a basis.

Think about it

Can three vectors of \(\mathbb{R}^2\) be independent?

No: there are more vectors than the dimension of the space, so the homogeneous system has more unknowns than equations and admits nontrivial solutions.

If \(\{u,v\}\) is independent, is \(\{u,v,u+v\}\)?

No: \(u+v\) is a linear combination of \(u\) and \(v\), so \(u+v-u-v=0\) is a nontrivial relation.

Exercises 10

  1. Are \((1,2)\) and \((2,4)\) linearly independent?
  2. Which of these subsets of \(\mathbb{R}^2\) is linearly independent?
  3. A set containing the zero vector is always:
  4. For \(v_1=(1,2,3)\) and \(v_2=(2,4,6)\), which coefficients \((c_1,c_2)\), not both zero, give \(c_1v_1+c_2v_2=0\)?
  5. For which value of \(k\) are \((1,2)\) and \((3,k)\) linearly dependent?
  6. The vectors \((1,0,1)\), \((0,1,1)\) and \((1,1,2)\) are:
  7. A set of 4 vectors in \(\mathbb{R}^3\) is always:
  8. For which value of \(k\) are \((1,0,2)\), \((0,1,1)\) and \((1,1,k)\) linearly dependent?
  9. If \(u\) and \(v\) are linearly independent, which of these sets is dependent?
  10. If \(v_1,\dots,v_k\) are linearly dependent, then:

What comes next

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §3.4
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §2.A
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §1.7