True or False on two statements.

a)Every independent set in $ℝ^n$ is orthogonal.
b)If {$x_1,x_2,...,x_n$} is orthogonal in $ℝ^n$, then $ℝ^n$=span {$x_1,x_2,...,x_n$}

My guess for a) is T because the independent vectors can be reduced to RREF and so they are orthogonal to each other, but I am not sure if I missed anything. For b) , I feel like it is a T but I don't know how to prove.

In fact I have poor idea how the orthogonality visualizes in 3-Dimension, someone help?

• Consider the vectors $(1~0)$ and $(1~1)$. – pjs36 Nov 22 '15 at 0:14
• Can you show that if $n$ vectors are mutually orthogonal, then they are linearly independent? – Théophile Nov 22 '15 at 0:19

a) False. Consider, as mentioned in the comments above the vectors $(1,0)$ and $(1,1)$. But more generally, think about the fact that given any two non-identical lines through the origin of $\mathbb{R}^2$, choosing a vector lying on each yields a basis for $\mathbb{R}^2$. The angle $\theta$ between these two vectors can range from $(0,\pi/2)$.