A Question about Span and Basis

[The Student]

My notes have, "If Span(X) = R^n, then X is a basis of R^n". How can that be true? A contradiction to the statement that I quoted: {(2,0), (1,0), (0,1)} spans R^2 but clearly not a basis for R^2. What am I missing?

 

[daon2]

My notes have, "If Span(X) = R^n, then X is a basis of R^n". How can that be true? A contradiction to the statement that I quoted: {(2,0), (1,0), (0,1)} spans R^2 but clearly not a basis for R^2. What am I missing?

You are correct. Even the more ridiculous example: \(\displaystyle \text{Span}(\mathbb{R}^n)=\mathbb{R}^n\).

If it was stated in class, perhaps your professor mentioned X was a linearly independent set? Or equivalently, if \(\displaystyle \text{dim}(V)=n, |X|=n\) and \(\displaystyle \text{Span}(X)=V\) then \(\displaystyle X\) is a basis for \(\displaystyle V\).

 

[The Student]

You are correct. Even the more ridiculous example: \(\displaystyle \text{Span}(\mathbb{R}^n)=\mathbb{R}^n\).

If it was stated in class, perhaps your professor mentioned X was a linearly independent set? Or equivalently, if \(\displaystyle \text{dim}(V)=n, |X|=n\) and \(\displaystyle \text{Span}(X)=V\) then \(\displaystyle X\) is a basis for \(\displaystyle V\).

Thanks a lot for clearing that up for me, I thinks it's just a flaw in the notes.