Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

Question

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to prove that a set of two vectors can't span a set of three vectors?

Answer 1

Hint; Let V be a vector space of dimension n . Then any basis of V contains n vectors. And these n vectors span V and are linearly independent. Do you know of any thereom that relates span and linear independence ?

Answer 2

First, know the definition of dimension and basis.

The dimension of a vector space is the number of vectors in any of its bases.

A basis for a vector space is a set of vectors that is linearly independent and spans the space.

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It is not solely about showing that a set of two vectors cannot span a set of three vectors, since this is possible if say a vector is a from the set of three vectors is a linear combination of the two vectors.

Instead, we have to take note of the definition given by a basis for a vector space. Reading closely, it is the maximal linearly independent set of vectors and the minimal spanning set of vectors for a vector space. Using these definitions properly, you can prove your problem via contradiction.