Linear Independence

We want to know how "big" of a piece of the full space a set of vectors would span. To do this, we say that a set of vectors \({v_1, v_2, \dots, v_n}\) is linearly independent if scalars \(c_1, c_2, c_3, \dots, c_n\):

\[ c_1v_1 + c_2v_2 + \cdots + c_nv_n = 0 \]

has only the trivial solution \(c_1=c_2=\cdots=c_n=0\).

This is because if it had nontrivial solutions, then one of the vectors could be written as a linear combination of the other vectors; i.e., the vectors become linearly dependent.

In this way, linear independence can be seen as measuring "redundancy": if one vector is a linear combination of the others, then it is redundant. To check for linear independence, we can simply take a matrix \(A\) whose columns are our set of vectors, and ask if the homogeneous system \(Ax=0\) has any nontrivial solutions.