Linear Independence

A solution set written as a span uses one spanning vector per free variable. Whether that spanning set is as small as possible, or carries a redundant vector that some combination of the others already reaches, is the question of linear independence. Like the questions before it, it reduces to counting pivots in a homogeneous system.

The definition

Independence is a property of a set of vectors, phrased through the homogeneous vector equation they generate.

The trivial solution always exists; independence says it is the only one. Dependence says there is a genuine one, a way to combine the vectors to zero without using all-zero weights.

Testing by row reduction

Because the test is whether a homogeneous vector equation has a nontrivial solution, it is a row reduction.

The augmented column of zeros never changes under row operations, so in practice one reduces $A=[{\mathbf{v}}_1\cdots {\mathbf{v}}_p]$ alone.

An independent set produces a pivot in every column.

Independence as a condition on a parameter

Many problems ask for the values of an unknown entry that make a set dependent. The dependence appears exactly when the reduction produces a free variable. Take

Here ${\mathbf{v}}_2=-3{\mathbf{v}}_1$ already, so $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ is dependent, and adding any ${\mathbf{v}}_3$ keeps the set dependent for every value of $h$: a dependence relation among the first two, padded with weight $0$ on ${\mathbf{v}}_3$, still has a nonzero weight. No value of the parameter can make an already-dependent set independent.

When the first vectors are independent, the parameter can matter. For

row reduction leaves a bottom-row pivot in column 3 unless $h$ takes one specific value that zeros it; at that value $x_3$ becomes free and the set turns dependent. The parameter is fixed by demanding the third pivot vanish.

Small cases by inspection

For one or two vectors, no computation is needed.

• One vector. $\{\mathbf{v}\}$ is independent if and only if $\mathbf{v}\ne 0$. The equation $x_1\mathbf{v}=0$ forces $x_1=0$ exactly when $\mathbf{v}$ is nonzero.
• Two vectors. $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ is dependent if and only if one is a scalar multiple of the other. Geometrically, dependent means the two lie on a common line through the origin.

Dependence means one vector is redundant

The name dependent is justified by rewriting a dependence relation.

If ${\mathbf{v}}_j=c_1{\mathbf{v}}_1+\cdots$ is a combination of the others, move it across: $(-1){\mathbf{v}}_j+c_1{\mathbf{v}}_1+\cdots =0$ is a dependence relation with nonzero weight $-1$. Conversely, given a dependence relation, let $j$ be the largest index with $c_j\ne 0$; then $c_j{\mathbf{v}}_j=-c_1{\mathbf{v}}_1-\cdots -c_{j-1}{\mathbf{v}}_{j-1}$ solves for ${\mathbf{v}}_j$ in terms of the earlier vectors.¹

This connects independence back to span. If $\mathbf{u},\mathbf{v}$ are independent in ${\mathbb{R}}^3$, they span a plane, and a third vector $\mathbf{w}$ makes $\{\mathbf{u},\mathbf{v},\mathbf{w}\}$ dependent exactly when $\mathbf{w}$ lies in that plane.

Two automatic dependences

Some sets are dependent for structural reasons, without any reduction.

The matrix $A=[{\mathbf{v}}_1\cdots {\mathbf{v}}_p]$ is $n\times p$, so $A\mathbf{x}=0$ has $n$ equations in $p>n$ unknowns; with more columns than rows, some column has no pivot, hence a free variable, hence a nontrivial solution.

If ${\mathbf{v}}_1=0$, the relation $1\cdot {\mathbf{v}}_1+0\cdot {\mathbf{v}}_2+\cdots +0\cdot {\mathbf{v}}_p=0$ has a nonzero weight.

For example, three vectors in ${\mathbb{R}}^2$ are always dependent (there are more vectors than entries), even when no one of them is a multiple of another — the dependence involves all three at once.

Pivot criteria

Independence, span, and the shape of solution sets all read off the pivot pattern. Collecting the criteria for the columns of an $m\times n$ matrix $A$:

Independence caps the number of vectors at $m$; spanning ${\mathbb{R}}^m$ requires at least $m$. A set that does both must have exactly $m=n$ vectors — the defining condition of a basis.