A calculus of the absurd

20.4.2 Linear independence

This is a \(\text {very important}^{TM}\) concept in linear algebra.

  • Definition 20.4.2 Let \(v_1, v_2, ..., v_n\) be some vectors in a vector space \(\textsf {V}\), and let \(a_1, a_2, ..., a_n\) be some scalars in the field \(\mathbb {K}\) (over which this vector space is defined).

    We say these vectors are linearly independent if and only if

    \begin{equation} a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0 \implies a_1 = a_2 = ... = a_n = 0. \end{equation}

    In words, this means “if the only values for all the \(a\)s which satisify \(a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0\) are when all the \(a\)s are zero, then the vectors are linearly independent”.