Linear independence

We say a set of vectors is linearly independent if the solution to the equation $c_1{\vec{u}}_1+\cdots +c_m{\vec{u}}_m=\vec{0}$ is the trivial solution $c_1=c_2=\cdots =c_m=0$. We can also express some of the vectors in the set as a linear combination of other vectors in the set.

We conversely say a set is linearly dependent if the above doesn’t hold true, i.e., if there is some solution that is not the trivial solution. Note the idea of redundancy as well — if there are redundant vectors we can get to the same place multiple different ways. A square matrix with linearly dependent columns is singular.

To solve for independence or dependence, we can load the vectors into a matrix and row reduce. A unique solution indicates linear independence. Otherwise the set will be linearly dependent. We can also compute the determinant. If $\text{det}(A)=0$, then the set is linearly dependent.

Functions

Linear combinations of functions can also be linearly independent. We them similarly to vectors where we have arbitrary constants that must be 0 to get 0 on the other side of the equation. In other words, we should not be able to express functions as combinations of each other.

A helpful tool, especially when considering solutions to ODEs, is the Wronskian determinant.

Especially useful for circuit analysis is the special case (which is pretty easy to see) for second-order ODEs. The solutions $x_1(t),x_2(t)$ are linearly independent if and only if we cannot write $x_1(t)=k_2{x}_2(t)$ or $x_2(t)=k_1{x}_1(t)$.