Suppose x, y are elements of R^n and are nonparallel vectors.
a) Prove that if sx + ty = 0, then s = t = 0.
b) Prove that if ax + by = cx + dy, then a = c and b = d.
a) Prove that if sx +ty = 0, then s = t = 0. If they are nonparallel than they can be chosen as the basis (orthogonal or not) in subset in Rn. We have ∑(i=1 to 2) aixi = sx +ty = 0. But these two vectors are linearly independent as they are two basis vectors and that mean that s = t = 0.
b) Prove that if ax + by = cx + dy, then a = c and b = d. We have ax + by = cx + dy → (a-c) x + (b-d) y = 0. Supposing the same as we write before we get a-c=0 b-d=0 a = c and b = d.