Prove that a set of $d+1$ vectors in $d$ dimensions is linearly dependent

Question

How would I prove that set of d+1 vectors in d dimensional space is linearly dependent? Could I use gaussian elimination and show that one of them is linearly dependent or does this follow from definition?

Answer 1

There are two cases to consider, depending on whether there is a linearly dependent proper subset. If there is, then the whole set is linearly dependent and you are done.

Suppose then that every proper subset is linearly independent. Take a subset of $d$ vectors, $\{v_1,...,v_d\}$. That subset is linearly independent, and it has the same cardinality as the dimension of the space, and so there is a theorem we can apply which concludes that $\{v_1,...,v_d\}$ is a basis (I make the assumption that you already have this theorem in your pocket). As a consequence, the subset $\{v_1,...,v_d\}$ spans.

Thus the $d+1^{\text{th}}$ vector is a linear combination of that subset of $d$ vectors: $$v_{d+1} = a_1 v_1 + ... + a_d v_d $$ and so $$a_1 v_1 + ... + a_d v_d - v_{d+1}=0 $$ showing that the whole set is linearly dependent.