Linear Independent Vectors

[jebz123]

If v1 = u1 + u2 + u3, v2 = u1 + a*u2, and v3 = u2 + b*u3...where u1, u2, and u3 are given linearly independent vectors, find the condition that must be satisfied by a and b in order to ensure that v1, v2, and v3 are linearly independent.

[Cap'n Refsmmat]

Okay. How would you start this? We're not going to just give you the answer.

 

How would you start, and where do you trip up?

[shif]

u1, u2 and u3 are the basis of the linear space they span (they are lin. indep. and span it by def.)

Write v1, v2 and v3 in a matrix according to the basis {u1,u2,u3} and you get

1 1 1

1 a 0

0 1 b

the will be lin. dep. if and only if we can make a zero row with Gauss elimination which is if and only if

b(a-1)+1=0 (if I am not mistaken)