What is the way of solving this question

[transgalactic]

what is the general way of solving this question:

http://img116.imageshack.us/img116/1152/25587465vv2.gif

??

 

[Defennder]

1. You want V(k) to be linearly independent so that it will form a basis for R3. So that means if you interpret them as row vectors in a matrix and perform row reduction, the end result should be a matrix whose rank is 3. So what does the matrix being of full rank imply? And what values of k are suitable such that the matrix is of full rank?

2. Firstly determine the solution space of U(k). Then span(U(k1)) \subseteq span(V(k2)) if \forall v \ \text{where v is a vector in basis of U(k1)} \ , v \in span(V(k2)). ie. try to express every vector in the basis of U(k1) as a linear combination of V(k2). Take note of when this is possible (ie. which values of k1 and k2 permit that).

 

[transgalactic]

i think i solved part 1.
but i don't know how to use part 1
in order to solve 2

http://img384.imageshack.us/img384/2546/55339538nk4.gif

 

[transgalactic]

what is a solution space?

i can find k values for which i get one solution
infinite solution
or no solution

what is the definition of solution space in a parameter matrix?

 

[transgalactic]

how do i find the solution vectors of U(k)?

 

[transgalactic]

when you say
"Firstly determine the solution space of U(k). "

there are 3 types of solution?(depends on the values of K)
no solution
1 solution
infinite solution

 

[transgalactic]

http://img408.imageshack.us/my.php?image=89390838kq8.gif