Question

Can a set of 5 vectors in R6 span all of R6?

Answer 1

To span R6 you need at least 6 vectors. So 5 vectors won't do.

Answer 2

Ok, that's what I thought, but I tend to second guess myself.
Thanks so much! :)

Answer 3

Cheers!

Answer 4

why is that there has to be 6 vectors to span R6, though?

Answer 5

It comes from the definition:

Let \(S = \{v_1, v_2, ...,v_n\}\) be a set of vectors in a vector space \(V\) and let \(W\) be the set of all linear combinations of those vectors. The set \(W\) is the span of the vectors and is denoted by \(W = \text{Span}(S)\).

Answer 6

The dimension of a vector space V is the number of vectors in a basis (the minimal spanning set). The dimension of R6 is 6, so there must be 6 vectors in the basis of R6. Any set that spans R6 will have at least 6 vectors, or possibly more (some may not be linearly independent).

Answer 7

ah, alright. thanks again :)