How do I find values that make up a desired linear combination of vectors?

I have 2 vectors A = [2 -1 2 1 0] B = [-1 1 1 0 1]

how can i get the 2 numbers x and y such that x*Ai + y*Bi gives me any vector C such that Ci > 0

[Dharshini]

use Matlab to determine if w=(2,7,-6) is a linerar combination of the vectors v1=(1,2,0), and v2=(0,-1,2).

Answer 1

[Roger Stafford]

There are infinitely many choices for x and y. Your inequalities can be reduced to:

0 < x < y < 2*x

Any pair of x and y satisfying these will cause C = x*A+y*B > 0.

[Bello Umar Muhamamd]

Thank you, but please how did you come up with this

[John D'Errico]

Simple algebra and some deductive thought will suffice.

A*x + B*y = [2*x - y, y - x, 2*x + y, x, y]

If you know that each of these elements MUST be positive, the you know that both x and y must be positive from the last two cases.

Since y-x must be greater than zero from the second, then you know that

0 < x < y

Since x and y are both forced to be positive, then 2*x+y is trivially positive.

Finally, you are given that 2*x-y>0. Therefore you know that 2*x>y.

So simple deductive reasoning yields the inequality chain:

0 < x < y < 2*x

Any values of x and y that satisfy the above will suffice. Again, there is no presumption that a solution always exists in general.

Answer 2

[John D'Errico]

Of course, there is no reason to know that any value of x and y actually exist that will satisfy the requirement. And even if you CAN find such a solution, then it would be of the form C(i) >= 0. Enforcing a strict inequality will be more problematic.

If you just want some solution though, this will suffice:

AB = [A',B']
AB =
     2    -1
    -1     1
     2     1
     1     0
     0     1

xy = linprog([1 1],-AB,-1.e-8*ones(5,1))
Optimization terminated.
xy =
   2.0084e-08
   3.0084e-08

A*xy(1) + B*xy(2)
ans =
   1.0084e-08        1e-08   7.0253e-08   2.0084e-08   3.0084e-08

[Bello Umar Muhamamd]

Thank you