The Student Room Group

Proof help pls

Can anyone check if my proof is correct?
(edited 3 years ago)
I'm finding it hard to read what you've written, in particular I can't make out the word after "when" on the 2nd line.

I'd say the "key" fact here is that the rank of the augmented matrix is less than or equal to the number of equations. I'm not convinced you've justified this sufficiently, but it may be fine.

You also have a logic error: you say "for the system to have an infinite number of solutions r < n", but what you actually need is "if r < n, the system has an infinite number of solutions".

In terms of layout / wording, I think your solution would be a lot clearer if you started off by actively defining m and n:

"Let m be the number of equations in the system, and let n be the number of variables. Then ..."
Original post by ottersandseals1
Apologies if it's hard to read. After when it says: considering the rank as the highest possible rank is that if it was equal to m. For a system to have an infinite number of systems r< n and so if we consider the fat that the max r is m then r <n.

For your second point, what would you suggest to fix it?

Okay, ill change that thanks

Will do thanks

It was just "considering" I couldn't make out (perhaps I should have been clearer and saved you a little writing). I don't want to give you too much of a hard time about your writing (and in this case it clearly wasn't helped by heavy indentations in the pad from whatever you wrote on a previous page), but for this particular piece of work, I'd say you're skirting that boundary where you might lose marks because an examiner literally can't read what you're trying to say (even after you've told me that's what that word says, I'm struggling to see it from what you've written).

For the logic error, you literally just need to write what I've said instead of what you said (note that this is also what the theorem said; in effect you somewhat had two logic errors, first that what you claimed was true wasn't what the theorem actually said, and secondly that what you claimed was true also didn't actually show what you wanted). Edit: this isn't right given the totality of what you wrote. Give me a moment...
(edited 3 years ago)
Original post by ottersandseals1
Apologies if it's hard to read. After when it says: considering the rank as the highest possible rank is that if it was equal to m. For a system to have an infinite number of systems r< n and so if we consider the fat that the max r is m then r <n.

For your second point, what would you suggest to fix it?

Okay, ill change that thanks

Will do thanks

Amended reply regarding your second point:

Looking more closely at what you've written, I think there's a general issue that needs addressing. When presenting a proof, you usually to start from the things you know are true, and end up with a conclusion that shows you've deduced the desired result. Instead, much of what you've done is start from what you want to be true and then tried to show those things actually are true. So, you've said that you want r < n, and then gone on to justify why that's true.

There are two issues with this:

(a) It's confusing (certainly for the reader, and often for the writer). Your argument goes "backwards and forwards" rather than going in a single direction from start to finish.
(b) You *can't* argue backwards (unless you know exactly what you are doing). The difference between A    B    C    DA \implies B \implies C \implies D v.s. A    B    C    DA \implies B \impliedby C \implies D". Is not just that one is easier to follow than the other, it's that the 2nd one doesn't actually show that A    DA \implies D. Explicitly here, arguing "I need r < n to prove my conclusion" is different from what you need, which is "Because I've shown that r < n, I can deduce my conclusion".

So you fundamentally need to reorder things so that you deduce r < n before talking about how many solutions there are.
Comments inline.

Original post by ottersandseals1
I've changed the paragraph to the following:

Let 'm' be the number of equations and let 'n' be the number of variables. The number of equations/variables of what? Notice how when I worded this, I explicitly said "of the system".

For a system of m equations has n variables. What is this supposed to mean? As written, it seems that you are saying "every system of m equations has n variables" which I very much doubt is what you meant.

This means m < n . Why? Why does it follow from the previous sentence?

We assume that the rank of the matrix is the highest possible matrix and so r = m. The rank of a matrix is a number - it can't equal "the highest possible matrix" (not that it's clear what you even mean by that). You haven't even defined what r means, so one certainly can't deduce that it equals m. And if your proof assumes something that isn't necessarily true, it's not a proof.

When we consider the fact that the maximum value of r is equal to m therefore we get r < n. Due to r < n we can deduce that the system has an infinite number of solutions. This is basically OK, but you should be explicit about the point where you are invoking the theorem you've been given.

Is this okay?
So, it's not great. However, although I've made a lot of comments, everything I've said is fairly easily fixable. In general, you're often using variables without defining them, or saying things that are fairly obviously not what you mean if you only read carefully over what you've said.

This probably all seems very frustrating right now, but it's largely a matter of practice, although you also a certain amount of conscious structuring of your argunent that you can largely manage without at A-level.
Original post by ottersandseals1
What should i change? At this point i don't know what else to change

You'll notice that most of my comments are in the form of questions. You need to answer them (if not to me, to yourself). For example, in the first sentence:

Let 'm' be the number of equations and let 'n' be the number of variables. The number of equations/variables of what?


The answer to the question is "the number of equations in the system". So you would rewrite this to say:

Let m be the number of equations and let n be the number of variables in the system.


If I ask you what something means, or why it's true, think about what you've written and ask yourself whether it's clear what it means, or why it's true. Then think about how you might want to change things to clarify matters.

Edit: And to be honest, there come's a point where you just need someone to answer it properly to show you. But that really needs to be a teacher on your course - if you just write down what I tell you then (a) it will make your teacher think you understand it better than you do, and (b) if you can't write your next assignment to that level (which is natural, since you're learning), it will be obvious you didn't do the first piece and you'll be in danger of getting a plagarism violation.
(edited 3 years ago)
Original post by ottersandseals1
Is this correct?

Let 'm' be the number of equations in the system, let 'n' be the number of variables in a consistent system and let 'r' be the rank of the augmented matrix.

A system of m equations that has n variables, means m < n . The rank of a matrix is the number of leading 1s in any row-echelon matrix to which it can be carried by row operations. The rank of a matrix will always be less than or is equal to the number of equations. We assume that the rank of the matrix in this case is the highest possible matrix and so this means the rank is equal to the number of equations, therefore r = m. When we consider the fact that the maximum value of r is equal to m therefore we get r < n as the number of variables will always be higher than the number of equations and this case we are assuming that r=m. A system will have an infinite number of solutions if r < n. Due to r < n we can deduce that the system has an infinite number of solutions.

I'd say there's no longer anything actively wrong. The whole "assume the rank of the matrix" is still bad, but probably not so bad you'd lose marks for it. There's a lot of unnecessary stuff.
Actually, thinking about it, this is still pretty bad. Here we go again:

Original post by ottersandseals1
Is this correct?

Let 'm' be the number of equations in the system, let 'n' be the number of variables in a consistent system and let 'r' be the rank of the augmented matrix. Why have you switched from the system to a? "The" implies you're talking about a particular system, "a" implies you're making a general statement about consistent systems.

A system of m equations that has n variables, means m < n . No it doesn't. x = 1, x = 1 is a system of 2 equations with 1 variable.

The rank of a matrix is the number of leading 1s in any row-echelon matrix to which it can be carried by row operations. This is true (well, I assume it is, it's about 30 years since I did this), but I'm not sure it's terribly relevant. If you omit this line, do you lose anything? Also, did you mean "carried" or something else?

The rank of a matrix will always be less than or is equal to the number of equations. What equations? Why are you talking about equations when you started off talking about matrices? Can you think of something more directly related to a matrix to talk about instead?

We assume that the rank of the matrix in this case is the highest possible matrix and so this means the rank is equal to the number of equations, therefore r = m. Why are you assuming this? You've previously said that r <= m, why do you think you need to assume r = m?

When we consider the fact that the maximum value of r is equal to m therefore we get r < n as the number of variables will always be higher than the number of equations and this case we are assuming that r=m.Why do you think the number of variables will always be higher than the number of equations? I've shown you this doesn't need to be true.

A system will have an infinite number of solutions if r < n. As I told you, you should explictly say something like: "By theorem 2.1.1 (whatever the actual number is), we know that a system will have ..."

Due to r < n we can deduce that the system has an infinite number of solutions.Again, does this really add anything to what you said in the line above? Or at least, can you very slightly reword the line above to make this line superfluous?

Bottom line on a lot of this: the way you use words and symbols matters. There are a lot of points in writing a mathematical proof where you need to be much more precise about your wording than you would in normal speech.

I'm also confused that I've told you to do specific things (e.g. reference the theorem at the point where you use it) and you haven't, and conversely I've told you not to do things (assume r = n) and you have.

Finally, I gave you a precise sentence to use for the beginning of your proof, and you decided you could word it better. Now, I'm not saying my wording is necessarily perfect, and there's nothing wrong with you trying to word it yourself. But under those circumstances you should certainly think carefully about anything you decide to change. I can't help feeling you didn't do so here.
Original post by ottersandseals1
Let 'm' be the number of equations, let 'n' be the number of variables, and let 'r' be the rank of the augmented matrix in a consistent system.

A system of m equations that has n variables, means m < n . The rank of a matrix is the number of leading 1s in any row-echelon matrix to which it can be carried by row operations. The rank of a matrix will always be less than or is equal to the number of equations. We assume that the rank of the matrix in this case is the highest possible matrix and so this means the rank is equal to the number of equations, therefore r = m. When we consider the fact that the maximum value of r is equal to m therefore we get r < n as the number of variables will always be higher than the number of equations and this case we are assuming that r=m. A system will have an infinite number of solutions if r < n which is highlighted by Theorem 1.2.2. Due to r < n we can deduce that the system has an infinite number of solutions.


I changed a bit. I don't how to say that r = m without explicitly saying so. Could you give me a model answer? I'm struggling cos I've got loads of deadlines

Frankly, it looks like you've ignored at least 80% of my comments. I think at this point you just need to accept this isn't going to be perfect and get some feedback from your teachers. You're in your first term - you're not expected to be writing model proofs off the bat.

The only other thing I'm going to say is that there really is an "accepted" way of writing mathematics. It feels like you feel compelled to "put things in your own words" (which might be expected in, say, an English class), but unless you really know what you're doing it's going to cause you problems in proof-writing.

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