Linear Independence of Vectors Spanning $\mathbb{R}^n$

[Dustinsfl]

If x₁, x₂,..., x_n span \mathbb{R}^n, then they are linearly independent.

This is true since n-1 vectors can't span R^n.

How can this be written in a more meaningful way?

 

[Squeezebox]

\mathbb{R}^n is an n-dimensional vector space. {x₁,x₂,...,x_n} is a spanning set of \mathbb{R}^n of length n. This makes {x₁,x₂,...,x_n} a basis of \mathbb{R}^n, which means it must be linearly independent.