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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?

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

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