Question #3204

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.

Expert's answer

a) Prove that if s**x** +t**y** = 0, then s = t = 0.

If they are nonparallel than they can be chosen as the basis (orthogonal or not) in subset in** R**^{n} .

We have

∑(i=1 to 2) a_{i} **x**_{i} = s**x** +t**y** = 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 a**x** + b**y** = c**x** + d**y**, then a = c and b = d.

We have

a**x **+ b**y **= c**x** + d**y** → (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.

If they are nonparallel than they can be chosen as the basis (orthogonal or not) in subset in

We have

∑(i=1 to 2) a

But these two vectors are linearly independent as they are two basis vectors and that mean that s = t = 0.

b) Prove that if a

We have

a

Supposing the same as we write before we get

a-c=0

b-d=0

a = c and b = d.

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