Question

Hi can anybody help me with this question? I've been trying it for well over two hours now and I'm no closer to solving it.

Find three vectors P, V, W ∈ R4 such that the solutions to the system
3w + 9x + 12y + 7z = 6
2w + 6x + 9y + 5z = 5
w + 3x + 12y + 5z = 10
have the form P + λV + μW with λ, μ ∈ R.

Answer 1

i don't get it sorry

Answer 2

Thank you for trying. :) Anyone else have any ideas?

Answer 3

Do you know how to solve the original system in some way?
My thoughts were using Gaussian Elimination to find a row echelon form, then we proceed from there. But perhaps you know a different method that you prefer.

A note is that we have four variables and only three equations, so we should have a free variable (one we can set to anything freely) to deal with.

Answer 4

row reducing will give 2 free variables, x_2, x_4

hint, the non constant vectors are [-3,1,0],0] and [0,0,-1/3,1]

Answer 5

Oh, you are correct. I hadn't tried it for myself beforehand. Thanks!

Answer 6

shuold say [-1,0,-1/3,1]

Answer 7

I didnt even see what you said:) im on a tablet and it makes it hard.

Answer 8

I had said there was only one free variable in my msg (looking at the equations vs. variables superficially). You said there were two free variables in the next reply, which I just checked and found for myself.. lol

Answer 9

I'm sorry but what do you mean by free variables? :/

Answer 10

Essentially, independent variables. You can choose their value arbitrarily and the other variables (dependent) will be based on those chosen values.

The relevance is that the system's solution has some degrees of freedom because we didn't give enough information to isolate a single solution. Four variables is generally solved for a point if we have four equations all linearly independent. In our case we only had three equations and it appears some linear dependence so that we have two variables that are considered 'free', or independently set.

That is why you are looking for the three vectors P + lambda V + mu W, there are those two degrees of freedom in having two free variables.

Answer 11

So what do I have to do first? Sorry if I'm being slow about this but zzr0ck3r didn't really explain..

Answer 12

So, I gotta go back to my question -- do you know Guassian elimination? Like, putting this into a matrix form and breaking it down into the reduced row echelon form?

Answer 13

Do you mean making it look something like
x x x x x
0 x x x x
0 0 x x x

Answer 14

A bit like that, yes. Although we usually want those first x's to be 1's
Like:
1 x x x | x
0 1 x x | x
0 0 1 x | x
Once we have that, we can work into the vector form.

Answer 15

Do those x's have to be 1's? I thought that this was what I had to do and no matter what row operations I did I couldn't get my matrix to look that way..

Answer 16

By those x's I mean the ones going along the diagonal.

Answer 17

I guess if you're more familiar with that way it should work. I usually see that we want a main diagonal of 1's though, but that is somewhat more for convenience. But you'll have more work in the end with the equations outside of matrices in the end to get it done I think.

Answer 18

What should really happen in our case is we'll end up with something like this:
x x x x | x
0 0 x x | x
0 0 0 0 | 0
That's where the second free variable is coming into play, because we lose the first and second columns trying to sort out the second and third rows.

Answer 19

But you should try it yourself to make sense of that part.

Answer 20

Okay thanks for your help :).

Answer 21

Glad to help! :)