[chaserelock]

One small note is that independent vectors are ALWAYS orthogonal in their space. That's by definition of linear independence. If they were not mutually orthogonal, then at minimum you could remove 1 vector from the set and still have the same space representation.

[lp251]

[1, 0] and [1, 1] are linearly independent but not mutually orthogonal. Unless I am understanding what you mean by "in their space".