Is the space spanned by a set of orthogonal vectors greater than the space spanned by the same amount of linearly independent vectors? Let's say that we have a set of $n$ vectors, each belonging to $R^m$ that do not form a basis for $R^m$. 
Intuitively, I feel like the space spanned by a set of orthogonal vectors will be greater than the space spanned by the same amount of linearly independent vectors. Now although this might now always be true, it intuitively feels like for the majority of cases this is true, as the volume of orthogonal vectors will be larger than the volume of linearly independent vectors (assuming magnitude if vectors are the same).
Can anyone shine some light on this intuition? 
 A: Not true. Suppose you are in the regular 3D space $(\Bbb R^3)$. Two linearly independent vectors will span a plane, independent if those two vectors are orthogonal. In fact, if you have a set of linearly independent vectors, you can use them to construct the same number of orthogonal vectors
A: In $\mathbb{R}^m$, no, the orthogonality doesn't matter. 
However, hold onto this idea for later. In infinite dimensions you can construct a sequence of vectors that get closer and closer to each other, while still being linearly independent, and then there are some technical senses in which you can say they span a smaller space than an orthonormal set. 
Also, in numerical linear algebra the "conditioning" of a set of vectors (a measure of how far they are from being orthonormal) is a really important concept. When a set of vectors is very ill-conditioned, although in principle it spans a given space, in practice it turns out that you can only effectively operate on a smaller space due to the amplification of numerical rounding errors.
