Linear Independence

Definition

Let $y_1,y_2,\dots ,y_n$ be $n$ vectors or functions

Linearly independent

If $\sum _{i=1}^n{k}_i{y}_i=0\Rightarrow k_1=k_2=\cdots =k_n=0$,
then $\{y_i|i=1,2,\dots ,n\}$ is linearly independent.

$y_n$ can’t be expressed using the others

Linearly dependent

If $\exists k_1,k_2,\dots ,k_n$, not all zero, s.t. $\sum _{i=1}^n{k}_i{y}_i=0$,
then $\{y_i|i=1,2,\dots ,n\}$ is linearly dependent.

Facts

Basis(solutions of the ODE) = linearly independent solutions

Mutually orthogonal $y_i\perp y_j(i\ne j)$ vectors $y_1,y_2,\dots ,y_n$ are linearly independent

Check of linear independence of a set of vectors

1. make a matrix using the given column vectors
2. apply Gauss Elimination to make an upper triangular or Echelon matrix
3. check the number of non-zero pivots