# A calculus of the absurd

##### 20.4.2 Linear independence

This is a $$\text {very important}^{TM}$$ concept in linear algebra.

• Definition 20.4.2 Let $$v_1, v_2, ..., v_n$$ be some vectors in a vector space $$\textsf {V}$$, and let $$a_1, a_2, ..., a_n$$ be some scalars in the field $$\mathbb {K}$$ (over which this vector space is defined).

We say these vectors are linearly independent if and only if

$$a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0 \implies a_1 = a_2 = ... = a_n = 0.$$

In words, this means “if the only values for all the $$a$$s which satisify $$a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0$$ are when all the $$a$$s are zero, then the vectors are linearly independent”.