Linear independence

Definition

A set of vectors $x_1\dots x_n$ are said to be linearly independent if there is no vector in the set that is a linear combination of the others.

$$\sum _{i=1}^n{a}_i{v}_i=0$$

System is linearly independent $⟺$ $\forall a_i=0$.

System if dependent if there exists a linear combination with coefficients different from 0.

Tip

Suppose that you have a matrix $Ax=b$ and you would like to see if $A$ is linearly dependent. Let $v_1\dots v_n$ be column vectors of $A$. Then $\sum c_i{v}_i$ yields $Ac=0$ where $c$ is the coefficients for each row vector. Then this matrix $A$ is linearly independent if $A$ doesn’t have any zero-rows, meaning that $\det A\ne 0$. One can say that here $A$ is the matrix for a linear system and each $v_i$ denote the coefficients of $x_i$ on all equations.

By using Wronskian

Theorem says that if the Wronskian of a set of functions is not zero even for one value of $t$ then the set of functions are linearly independent.