Download Problem Suppose that u and v are two vectors in R n. Let a = u + v b

Problem Suppose that ~u and ~v are two vectors in Rn . Let
~a = ~u + ~v ~b = 2~u + 3~v ~c = 4~u + 7~v .
Find a linear relation between ~a, ~b and ~c. Explicitly, this means to find scalars p, q
and r, not all zero, such that p~a + q~b + r~c = 0.
Solution: Let’s see what relations we would need p, q and r to obey. We want
p~a + q~b + r~c = 0
Plugging in the expressions we have for (a, b, c), we get
p(~u + ~v ) + q(2~u + 3~v ) + r(4~u + 7~v ) = 0
Expanding and regrouping, we want
(p + 2q + 4r)~u + (p + 3q + 7r)~v = 0.
The way to guarantee that happens is to find (p, q, r) so that
p + 2q + 4r = 0
and p + 3q + 7r = 0
We can solve these linear equations in the standard manner. One solution is (p, q, r) = (2, −3, 1).