Linear Independence

Independent vectors contribute genuinely new directions; dependent ones hide a linear relation.

The idea

Linear independence is the test for whether a list of vectors contains a redundant one. A vector that can be built from the others in its list reaches nothing they do not already reach, so removing it changes nothing about what the list can describe.

One computation tests the whole list at once. Consider the ways of scaling the vectors and adding the results to produce the zero vector. Taking every coefficient to be $0$ always works. If that is the only way, the vectors $v_1, \dots, v_k$ are linearly independent: $c_1 v_1 + \cdots + c_k v_k = 0 \quad \text{forces} \quad c_1 = \cdots = c_k = 0.$ Otherwise they are linearly dependent, and any relation with a nonzero coefficient exhibits the redundancy: solve the relation for the vector carrying that coefficient, and it is written in terms of the others. The figure shows an independent pair in the plane: neither of $v_1$, $v_2$ is a scalar multiple of the other, so only the trivial combination reaches $0$.

The surrounding space caps how long an independent list can be. In $\mathbb{R}^{n}$ there are lists of $n$ independent vectors, and never a list of $n + 1$.

Ways to work on it

• Walkthrough. The definition, the trivial solution, and why three vectors in the plane must depend.
• Practice. Decide whether pairs of vectors in the plane are independent.
• Hardest. Produce an explicit nontrivial dependence relation.

Not sure where to start? Take the ten-question placement test.