Linear spaces: Given group of matrices, is it a field w.r.t addition, multiplication

yossa

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Given group K, defined as:

. . . . .\(\displaystyle K\, =\, \left\{\left(\begin{array}{cc}a&2b\\b&a\end{array}\right)\, \big|\, a,\, b\, \in\, \mathbb{Q}\right\}\)

Is the group a field in relation to the addition and multiplication of the matrices?

Thanks :)
 

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Does every nonzero matrix have a multiplicative inverse?
Hmm, I wonder. Try to find the inverse of (1 2 2 4) and then you tell us what you get. (for the record, row 1 is 1 2 and row 2 is 2 4)
 
OP, I suggest you show any axioms for multiplication are true, or perhaps find one that isn't along the way. Axioms relating only to addition are free, since you know it is a commutative group already. Associativity under multiplication is also true in general for matrices.

Hmm, I wonder. Try to find the inverse of (1 2 2 4) and then you tell us what you get. (for the record, row 1 is 1 2 and row 2 is 2 4)

I misread the picture, but this does not fit the form of OP's question either
 
I really do not succeed, can someone please show me the real solution?
thanks
 
I really do not succeed, can someone please show me the real solution?
thanks

You'll be far more likely to get the help you want if you start showing some work. That is the way to start.

List those "7 features" you mentioned.

Then express each of them in terms of the particular set you are working with. For example, if x and y are in set k, they can be expressed as ((a 2b)(b a)) and ((c 2d)(d c)), where a, b, c, and d are all rational numbers. What does the closure of multiplication mean for these two matrices? And so on.

If you can at least state what has to be proved, and tell us which of those you have done, we can help you work out the rest. But you must show us what parts you are able to do, so we have a place to start. Read the Read Before Posting announcement, if you haven't already.
 
You'll be far more likely to get the help you want if you start showing some work. That is the way to start.

List those "7 features" you mentioned.

Then express each of them in terms of the particular set you are working with. For example, if x and y are in set k, they can be expressed as ((a 2b)(b a)) and ((c 2d)(d c)), where a, b, c, and d are all rational numbers. What does the closure of multiplication mean for these two matrices? And so on.

If you can at least state what has to be proved, and tell us which of those you have done, we can help you work out the rest. But you must show us what parts you are able to do, so we have a place to start. Read the Read Before Posting announcement, if you haven't already.

hey,
I want to start but I do not understand how I get a group out of this matrix ....
 
hey,
I want to start but I do not understand how I get a group out of this matrix ....

That's a start -- until now, you haven't told us anything about what you don't understand, so we haven't had any way to help. (Even telling you the whole answer probably wouldn't have given you the understanding you need.)

Maybe you just don't get what the whole problem is asking.

Let's start with the fact that K has to be a group under addition. That means that the sum of any two elements of K must be in K, and that addition must be associative and have an (additive) inverse, right? The latter two are true of any matrices, but you'll have to show that the additive inverse is in the set K.

So what about closure? That means that for any two elements x = ((a, 2b),(b, a)) and y = ((c, 2d),(d, c)), x + y = ((a+c, 2b+2d),(b+d, a+c)) must be able to be written in the same form. Can it? Yes: x + y = ((a+c, 2(b+d)),(b+d, a+c)) = ((m, 2n),(n, m)) where m = a+c and n = b+d. So x + y is in K.

Now do the same sort of thing to show that -x is in K. And so on.

One more thing: does your textbook have any examples of showing that a set is a field? Have you looked through that carefully to see the kinds of thinking that are involved? (Also, I did ask you to state the 7 properties you said you have to prove; please do so, because they might be stated in a different way than I assume. It's important to cooperate with people you ask for help.)
 
That's a start -- until now, you haven't told us anything about what you don't understand, so we haven't had any way to help. (Even telling you the whole answer probably wouldn't have given you the understanding you need.)

Maybe you just don't get what the whole problem is asking.

Let's start with the fact that K has to be a group under addition. That means that the sum of any two elements of K must be in K, and that addition must be associative and have an (additive) inverse, right? The latter two are true of any matrices, but you'll have to show that the additive inverse is in the set K.

So what about closure? That means that for any two elements x = ((a, 2b),(b, a)) and y = ((c, 2d),(d, c)), x + y = ((a+c, 2b+2d),(b+d, a+c)) must be able to be written in the same form. Can it? Yes: x + y = ((a+c, 2(b+d)),(b+d, a+c)) = ((m, 2n),(n, m)) where m = a+c and n = b+d. So x + y is in K.

Now do the same sort of thing to show that -x is in K. And so on.

One more thing: does your textbook have any examples of showing that a set is a field? Have you looked through that carefully to see the kinds of thinking that are involved? (Also, I did ask you to state the 7 properties you said you have to prove; please do so, because they might be stated in a different way than I assume. It's important to cooperate with people you ask for help.)


euso First of all thanks a lot It's not clear what, I'm not sending anything because I have not started doing the exercise, just now I'll start it with your help.
The 6 characteristics that I spoke to prove that:
1. The group is closed.
2. Collective and multiplication operation.
3. The addition and multiplication operation.
4. Q has a neutral organ.
5. All Q parts are reversible.
6. Breakdown.


Thanks sir.
 
euso First of all thanks a lot It's not clear what, I'm not sending anything because I have not started doing the exercise, just now I'll start it with your help.
The 6 characteristics that I spoke to prove that:
1. The group is closed.
2. Collective and multiplication operation.
3. The addition and multiplication operation.
4. Q has a neutral organ.
5. All Q parts are reversible.
6. Breakdown.

Those are not at all what I expected to see, though I can see how most are related to the list I had in mind.

Can you give fuller explanations of each? What is meant by "collective", "neutral organ", and "breakdown", in particular?

Then start showing some work.
 
The 6 characteristics that I spoke to prove that:
1. The group is closed.
2. Collective and multiplication operation.
3. The addition and multiplication operation.
4. Q has a neutral organ.
5. All Q parts are reversible.
6. Breakdown.

Having looked again, I suspect that these are attempts to translate from another language the same facts I was expecting. Please confirm that you mean this:

1. The group is closed under addition and under multiplication.
2. Addition and multiplication are associative.
3. Addition and multiplication are commutative.
4. The set has a "neutral element" (identity element) for addition, and for multiplication.
5. Every element has an additive inverse, and every non-zero element has a multiplicative inverse.
6. Multiplication is distributive over addition.

Your summary, even after replacing odd words with those commonly used, is lacking in important details.

But now you must apply this. Try first, as I suggested, to determine whether your set is closed under each operation. Some of these facts, as has been mentioned, are "inherited" from the fact that the elements are matrices, and don't need a separate proof. The hardest may be that the multiplicative inverse is in set K.
 
Those are not at all what I expected to see, though I can see how most are related to the list I had in mind.

Can you give fuller explanations of each? What is meant by "collective", "neutral organ", and "breakdown", in particular?

Then start showing some work.

Dear sir, with your help and help drop from the book I was able to solve this habit :)
Thanks:)
 
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