
# 9.1: Linear Independence

One of the core concepts in linear algebra is linear independence, and this concept translates to general vector spaces with no difficulty.

Definition 9.1.0

Let $$S$$ be a set with elements $$s_i$$. A linear combination of elements $$\{s_1, s_2, \ldots, s_n\}$$ is given by any finite sum $$\sum_{s\in S}c_s s$$ with coefficients $$c_s\in k$$. (If $$S$$ is an infinite set, then all but finitely many $$c_s$$ must be equal to $$0$$.)

Definition 9.1.1

Let $$S$$ be a set of vectors in a vector space $$V$$. Then we say that $$S$$ is linearly dependent if there exists a linear combination of elements of $$S$$ equal to $$0$$.

Example 9.1.1

Let $$\mathbb{R}^\infty$$ be the vector space of sequences of elements of $$\mathbb{R}$$. (ie, the space of sequences $$r=(r_1,r_2, r_3, \ldots)$$, with coordinate-wise addition and the usual scalar multiplication.) Let $$r_i\in \mathbb{R}^\infty$$ be the sequence with $$(e_i)_i=1$$ and $$(e_i)_j=0$$ for all $$j\neq i$$. Let $$n$$ be the element $$(-1, -1, -1, \ldots)$$. Now, let $$S$$ be the set of all the $$e_i$$ and $$n$$. This is actually a linearly independent set. You might note that the sum of all of the elements in $$S$$ (with all coefficients in the sum equal to $$1$$) seems to be the $$0$$-vector. But this is an infinite sum, and is thus not considered a linear combination of elements of $$S$$.

### Contributors

• Tom Denton (Fields Institute/York University in Toronto)