Find vectors u and v

Question

Let W be the set of all vectors of the form [2s-4t,-2s,-2s+t,3s-3t]

Find vectors u and v such that

W=span {u,v}

u=?

v=?

Answer 1

Dalia,

 

Two independent vectors U & V will span the set W if you have constants s & t such that:

sU + tV = member of W.

 

Well lo and behold the general vector in the set W already has s & t constants in its components. So looking at the pattern there, let's try all the coefficients in the s terms for U and the t terms for V:

U = (+2, -2, -2, +3)

V = (- 4,  0, +1, -3)

Then    s(+2, -2, -2, +3) + t(-4, 0, +1, -3) = (2s-4t, -2s, -2s +t, 3s - 3t) or the general vector in set W.

 

So it all works