Answer to Question #3204 in Linear Algebra for kalong08

Question

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.

Answer 1

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) a_i x_i = 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.