I will show my work and hopefully someone will be able to help me with the last part.
a) If are independent that means that . Now replace the vectors with their values and you get: . Now open parenthesis and then have the vectors on the outside: . It is given that is a basis for . That means that are all independent vectors. Since we know they are independent, we also know that the numbers inside the parenthesis = 0
and the matrix look as follows:
Then we get to this matrix:
Now, the only way the vectors can be dependent is if it is possible to have 2 rows of zero vectors. In a case where both the bottom rows equal 0!
So, vectors are independent when k does not equal 2
vectors are dependent when
b) When are dependent:
Now, like before I replace the v vectors with their u vector values: and open parenthesis and then put the u vectors on the outside: And just like before, since we know that are independent:
and from this we get that and and that means:
c) This is where I am unsure. The task is to find the value of k which would make span . The rule is that you need at least 4 independent vectors to span . The group they gave us contains 5 vectors and we know the the u vectors are independent. So now I need to prove that at least one of the v vectors are independent.
So let's take . If is not a linear combination of the group then we know that the group spans . But let's assume it does and check: Now replace the v vectors with their u vector values... : and do what we did before to get: and since we know that
is independent... :
and since -1 does not equal 0 that must mean that is not a linear combination of the group which means that the group spans for ANY VALUE OF K. Can someone please tell me if I came to the write conclusion?? This is pretty urgent. I have an assignment do on Friday so I do not have a lot of time.
Thanx so much.