Hi, I am new with linear algebra, and I'm having a hard time wrapping my mind around the 0 vector and the additive identity v + 0 = v, where 0 is the 0 vector. If I had a 2x2 matrix, and v + w = C + (C^T)*D ... (where (C^T) is the transpose, v & w are vectors, and C & D are matrices)... would the additive identity hold? I feel like it wouldn't, because I don't see how it would be unique... but I think I may be wrong.. can someone please help?
I can't make sense of your expression, how does adding two vectors get you a 2x2 matrix? Are you confusing something here, or am I the confused one? Anyway, if the additive identity does not hold, you're not dealing with a vector space and all bets are off (as far as linear algebra is concerned). One of the requirements of a vector space [tex]V[/tex] is that there exists an element [tex]\mathbf{0} \in V[/tex] such that [tex]\mathbf{v} + \mathbf{0} = \mathbf{v}[/tex] for all [tex]\mathbf{v} \in V[/tex].
That's what I'm trying to prove though, that the additive identity v + 0 = v does in fact hold, and if not it's not a vector space, but we have to test the axioms anyway to see which ones do hold. Addition of 2 vectors in this problem translates to: vector v := C (where C is a 2x2 matrix) vector w:= D (where D is a 2x2 matix) (v+w):= C + (C^T)D so i would set up my equation as v + 0 =? v C + (C^T)D =? C
If 0 is a 2x2 zero-matrix (the 0-vector you were referring to), then v + 0 = v, but 0 + v = 0, which can't hold for a vector space.