Linear Algebra/Definition and Examples of Linear Independence

Spanning Sets and Linear Independence

We first characterize when a vector can be removed from a set without changing the span of that set.

The lemma says that if we have a spanning set then we can remove a $\overrightarrow{v}$ to get a new set $S$ with the same span if and only if $\overrightarrow{v}$ is a linear combination of vectors from $S$. Thus, under the second sense described above, a spanning set is minimal if and only if it contains no vectors that are linear combinations of the others in that set. We have a term for this important property.

Here is an important observation:

${\overrightarrow{s}}_0=c_1{\overrightarrow{s}}_1+c_2{\overrightarrow{s}}_2+\cdots +c_n{\overrightarrow{s}}_n$

although this way of writing one vector as a combination of the others visually sets ${\overrightarrow{s}}_0$ off from the other vectors, algebraically there is nothing special in that equation about ${\overrightarrow{s}}_0$. For any ${\overrightarrow{s}}_i$ with a coefficient $c_i$ that is nonzero, we can rewrite the relationship to set off ${\overrightarrow{s}}_i$.

${\overrightarrow{s}}_i=(1/c_i){\overrightarrow{s}}_0+(-c_1/c_i){\overrightarrow{s}}_1+\dots +(-c_n/c_i){\overrightarrow{s}}_n$

Template:AnchorWhen we don't want to single out any vector by writing it alone on one side of the equation we will instead say that ${\overrightarrow{s}}_0,{\overrightarrow{s}}_1,\dots ,{\overrightarrow{s}}_n$ are in a linear relationship and write the relationship with all of the vectors on the same side. The next result rephrases the linear independence definition in this style. It gives what is usually the easiest way to compute whether a finite set is dependent or independent.

The above examples, especially Example 1.5, underline the discussion that begins this section. The next result says that given a finite set, we can produce a linearly independent subset by discarding what Remark 1.6 calls "repeats".

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Linear Independence and Subset Relations

Theorem 1.12 describes producing a linearly independent set by shrinking, that is, by taking subsets. We finish this subsection by considering how linear independence and dependence, which are properties of sets, interact with the subset relation between sets.

Restated, independence is preserved by subset and dependence is preserved by superset.

Those are two of the four possible cases of interaction that we can consider. The third case, whether linear dependence is preserved by the subset operation, is covered by Example 1.13, which gives a linearly dependent set $S$ with a subset $S_1$ that is linearly dependent and another subset $S_2$ that is linearly independent.

That leaves one case, whether linear independence is preserved by superset. The next example shows what can happen.

So, in general, a linearly independent set may have a superset that is dependent. And, in general, a linearly independent set may have a superset that is independent. We can characterize when the superset is one and when it is the other.

(Compare this result with Lemma 1.1. Both say, roughly, that $\overrightarrow{v}$ is a "repeat" if it is in the span of $S$. However, note the additional hypothesis here of linear independence.)

Lemma 1.16 can be restated in terms of independence instead of dependence: if $S$ is linearly independent and $\overrightarrow{v}\in ̸S$ then the set $S\cup \{\overrightarrow{v}\}$ is also linearly independent if and only if $\overrightarrow{v}\in ̸[S].$ Applying Lemma 1.1, we conclude that if $S$ is linearly independent and $\overrightarrow{v}\in ̸S$ then $S\cup \{\overrightarrow{v}\}$ is also linearly independent if and only if $[S\cup \{\overrightarrow{v}\}]\ne [S]$. Briefly, when passing from $S$ to a superset $S_1$, to preserve linear independence we must expand the span $[S_1]\supset [S]$.

Example 1.15 shows that some linearly independent sets are maximal— have as many elements as possible— in that they have no supersets that are linearly independent. By the prior paragraph, a linearly independent sets is maximal if and only if it spans the entire space, because then no vector exists that is not already in the span.

This table summarizes the interaction between the properties of independence and dependence and the relations of subset and superset.

	$S_1\subset S$	$S_1\supset S$
$S$ independent	$S_1$ must be independent $S_1$ may be either $S_1$ may be either $S_1$ must be dependent
$S$ dependent

In developing this table we've uncovered an intimate relationship between linear independence and span. Complementing the fact that a spanning set is minimal if and only if it is linearly independent, a linearly independent set is maximal if and only if it spans the space.

In summary, we have introduced the definition of linear independence to formalize the idea of the minimality of a spanning set. We have developed some properties of this idea. The most important is Lemma 1.16, which tells us that a linearly independent set is maximal when it spans the space.

Exercises

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