This is for a homework I had to submit yesterday. I think I managed to answer all the questions correctly, except the last one.
I had the following U set and v vector:
I had to prove that U is a subspace for V³, which I did.
Then I found:
Then I was asked to find a base for this set, and I found:
but I could also just have used the U specified above as a base. (I think)
I was then asked to find a base for , I got:
found previously also belongs to .
Finally, I was asked to say if belonged to U, and if not to express in function of and :
I plugged the values of v in and it gave me 21, which is not equal to 0, so I said it wasn't in U. That's when my problems started, I was never able to express in function of and !
Which gave me 3 equations and I tried to solve for m and n, but I wasn't able to. Now I know I have two orthogonal vectors (so they form a plane) and I know I need to use them to describe a third vector, so this third vector HAS to be on that plane too (that's why it doesn't work above). The problem I'm stuck with is that I don't know how to find and that also form a plane in which is!
Thanks in advance for helping me out.