linearly independent vectors

Topics

• linear algebra

A set of vectors $\{v_1,v_2\dots v_n\}\in \mathbb{V}$, where $\mathbb{V}$ is a vector space, is linearly independent, if the only way to make their linear combination equal the 0 vector, is by having every coefficient be 0. Mathematically:

$$\sum _{i=1}^n{\alpha }_i{v}_i=0⟹{\alpha }_1={\alpha }_2=\cdots ={\alpha }_n=0$$

Think of it this way, if we have 3 vectors $v_1,v_2$ and $v_3$, with $v_2=3v_1$ (i.e not linearly independent), then to form the 0 vector, we can have coefficients as $\{-3,1,0\}$ respectively, which are not all zeros.

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