ottersandseals1
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This is Q
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DFranklin
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This is a "just do it" question - it's hard to give much of a hint.

But for a first step: try to solve (find a matrix that works) for the case where x = (1,0,0,...0); i.e. 1 in the first component, all other components 0.

From there, try to solve for a general x.
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DFranklin
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(Original post by ottersandseals1)
Something like this? For the first step?
No. How can this work for an arbitrary vector y? You don't even *mention* y...
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DFranklin
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(Original post by ottersandseals1)
Is this on the right track?
Well, kind of but not really. Try the same approach, but with the assumption on x that I gave you. Note that at some point you'll need to actually make some conclusions: you want to be able to say "this is what a_{11} should equal, this is what a_{12} should equal, and so on".

Also note that what ever you do has to be valid for \mathbb{R}^n, but you are using 2x2 matrices so what you're writing would only be valid in \mathbb{R}^2.
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DFranklin
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Better, but (a) it doesn't actually work due to algebra errors, (b) what happens if you're dividing by zero, and (c) you will at some point need to generalise to R^n.
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DFranklin
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(Original post by ottersandseals1)
Aw yeah, if i divide by 0 it will be zero. So it doesn't work. I'm not sure where to go from here
If x isn't zero, there will be *some* component that isn't 0.
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DFranklin
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(Original post by ottersandseals1)
Could you give me an idea of how to start it? and then i could try finish it?
If you fix your previous answer, you'll see it crucially depends on a particular component of x not being 0.

You'll then have to work out how if I told you "only the kth component of my vector is non-xero" you could make a similar solution that only divides by x_k.
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DFranklin
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Getting there: that works if x1 is non-zero. What if x1 is zero but x2 is non-zero? What do you need to change?

Also, don't forget that you are going to need to provide a general solution at some point, so you'll need to think about what you'd do for a general n x n matrix and vector {\bf x} = (x_1, x_2, ..., x_n).
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DFranklin
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(Original post by ottersandseals1)
Wouldn't x2 and x3 always be multiplied by 0 so wouldn't make a difference? I'm stuck with how to turn this to a general solution
Think about why your solution only depends on x1?
How would you change it to only depend on x2?

If you really can't work it out, start from a general 3 x 3 matrix A, and assume x = (0, 1, 0), multiply through, and see what you get, and therefore what you'd need to do to force Ax = y.
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DFranklin
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(Original post by ottersandseals1)
i would have a non-trivial column 2?
Yes.

So you should now be in a position to write your proof. As a hint, it should probably start with a line something like:

Suppose {\bf x} = {x_1, x_2, ..., x_n}, {\bf y} = {y_1, y_2, ..., y_n} and x is non-zero. Then we can find k such that x_k \neq 0. Then...
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DFranklin
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(Original post by ottersandseals1)
so do i write three matrices with non-trivial solutions in different columns?
See above (I edited while you were replying). But, (and this is something I've said about 4 times now): you can't assume you're dealing with 3x3 matrices, so you will have to word a solution that works for all possible n x n matrices.

It's perfectly fine to say things like "A is the matrix with all columns zero, except..." however.
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DFranklin
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(Original post by ottersandseals1)
Ive added this. I've still to add the line u mentioned above.
I don't understand what you've written. You seem to be dividing by x1,x2,...xn, so now your solution doesn't just depend on x1 being non-xero, but also x2,x3,... In other words, you've made things worse, not better.
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DFranklin
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(Original post by ottersandseals1)
Damn, could you possibly write the solution to the end of it so i can try understand it? It would help immensely
Full solutions aren't allowed here. At the end of the day, I give people hints so they can solve a problem themselves. When it becomes me basically answering the question for them, I think they are better off asking their teacher/lecturer.
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