Prove: any k + 1 k-vectors are linearly dependent. (You can do it by using mathematical induction.)
Suppose we have k+1 k-vectors. Each such vector has k components. We assume as an induction hypothesis that any k (k-1)-vectors are linearly dependent, which means that there must be a linear dependence among any k (k-1)-component vectors.
(This hypothesis is trivial for k = 2)
We notice that a linear combination of linear combinations is a linear combination. This means if we can take our k+1 k-vectors and produce k linear distinct linear combinations of them that are each (k-1)-vectors, we can use the induction hypothesis to give us a linear dependence among these which will produce a linear dependence among our original vectors.
Notice also that k-vectors all of whose last components are 0 can be considered to be k-1-vectors.
So we pick one of our vectors, say the k-th, which has a non-zero k-th component. (if there is no vector with non-zero k-th component, then we really have k-1-component vectors and can apply the induction hypothesis immediately) Now we subtract enough of this vector from each of the others to make the resulting k-th components all zero.
Now we have our k (k-1)-vectors and find a linear combination that is 0. This gives a linear combination of the original vectors which is 0 and we are done.
We have to verify that the new linear combination cannot have all 0 coefficients
if the one obtained from the induction hypothesis did not. This is obvious because
each of our vectors except for the k-th occurs in exactly one of the combinations
to which the induction hypothesis was applied. Any non-zero coefficient in the
linear combination of linear combinations will give rise to a non-zero coefficient
of the corresponding one of our original vectors and we are really done.