Question

Let V be a vector subspace of R^n for some n?N. Show that if k>dim(V) then the set of any k vectors in V is dependent.

Answer 1

We know that dimension of a vector space is defined to be the cardinality of (ie, number of items in) its basis.

We also know that every linearly independent set can be extended to a basis by appending some or no vectors to the set. (Theorem)

Therefore, since size of the independent set (let's call this set as S) is k and it is strictly greater than V, we can write k = dim(V) + n1 (where n1 is a positive integer). From the theorem stated above, we know that there exist D vectors (D being a whole number) which when appended to set S makes it a basis (say B)

then since by definition of dimension of vector space, dim(V) = size(B) = dim(V)+n1+D

which would imply n1+D = 0, implying n1 = 0 which is a contradiction. Therefore the supposition that more than dim(V) vectors can be independent in a dim(V) dimensional space is contradictory.

Proof Completes here :)