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?
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 $(0,\pi/2)$.
b) True. If a list of vectors are pairwise orthogonal, they are necessarily linearly independent. This follows from some definitions and properties of the inner product (or in this case the dot product).
EDIT: Consider this helpful post. "Orthogonality and linear independence"