Need help, URGENT maths question: (1 Viewer)

1008

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In other words, the first part of the question follows due to the familiar properties of arithmetic in fields like the real/complex numbers.

(Also, your sums have been referring to the i-j components of matrices rather than the matrices themselves, so it wouldn't be right to say A(lambda B) equals those, for instance.)
Thanks! Yeah, just realised that I violated order of operations lol. x I'll get back to figuring out the second part of (1). Would this be right for (3) though?


EDIT: the first step is meant to have brackets around the whole thing, i.e.
 
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InteGrand

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Thanks! Yeah, just realised that I violated order of operations lol. x I'll get back to figuring out the second part of (1). Would this be right for (3) though?


EDIT: the first step is meant to have brackets around the whole thing, i.e.
Yeah that's correct, except make sure to put []ij around the matrices, since those sums aren't equal to matrices, they're equal to entries of matrices. So what you've shown is that the ij entry of A(B+C) is equal to the ij entry of (AB + AC), so the matrices are equal since all their entries are equal.

(Actually, you showed that the ij entry of A(B+C) equals [AB]ij + [AC]ij, but from the definition of matrix addition, this is just [AB+AC]ij.)
 

1008

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Yeah that's correct, except make sure to put []ij around the matrices, since those sums aren't equal to matrices, they're equal to entries of matrices. So what you've shown is that the ij entry of A(B+C) is equal to the ij entry of (AB + AC), so the matrices are equal since all their entries are equal.

(Actually, you showed that the ij entry of A(B+C) equals [AB]ij + [AC]ij, but from the definition of matrix addition, this is just [AB+AC]ij.)
Right, is this better?



Also, do I need to give reasons at each step like you do?
 
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InteGrand

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Right, is this better?



Also, do I need to give reasons at each step like you do?
I'm not sure about the necessity of giving reasons for each step (I mainly gave them for your own benefit). But it would probably be good to mention why the last line at least is true (reason being that the second last line is [AB]ij + [AC]ij, which is [AB + AC]ij by definition of matrix addition).
 

1008

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I'm not sure about the necessity of giving reasons for each step (I mainly gave them for your own benefit). But it would probably be good to mention why the last line at least is true (reason being that the second last line is [AB]ij + [AC]ij, which is [AB + AC]ij by definition of matrix addition).
Do you mean
"[AC]ij + [BC]ij, which is [AC + BC]ij" or am I just hallucinating?
 

InteGrand

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Do you mean
"[AC]ij + [BC]ij, which is [AC + BC]ij" or am I just hallucinating?
Oh yeah you're right, sorry, forgot the Q. was right distributivity rather than left distributivity.
 

InteGrand

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Wow this contradicts with the UNSW notes, which calls this the left distributive law, and the other one the Right distributive law???
I don't know what those notes say, but if they call them the other way round, then yes, there is a contradiction. You can search up online for other references to left- and right- distributivity. For example, here is a page on Proof Wiki where they prove a left distributivity thing, and you can see it matches the Wikipedia definition: https://proofwiki.org/wiki/Left_Distributive_Law_for_Natural_Numbers .

Wolfram says the same thing: http://mathworld.wolfram.com/Distributive.html .
 

1008

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I don't know what those notes say, but if they call them the other way round, then yes, there is a contradiction. You can search up online for other references to left- and right- distributivity. For example, here is a page on Proof Wiki where they prove a left distributivity thing, and you can see it matches the Wikipedia definition: https://proofwiki.org/wiki/Left_Distributive_Law_for_Natural_Numbers .

Wolfram says the same thing: http://mathworld.wolfram.com/Distributive.html .
Right thanks, I'll have to clarify this with my lecturer lol. Anyway, for (4), here's my working, if this is right?


Don't get how to do (3) tho...Can you still do it with this notation and how would you generalise that to A of any size and the square matrix I of any size?

Just for reference:
(3) prove AI = A and IA = A, where I represents identity matrices of the appropriate (possibly different) sizes
 
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InteGrand

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Right thanks, I'll have to clarify this with my lecturer lol. Anyway, for (4), here's my working, if this is right?


Don't get how to do (3) tho...Can you still do it with this notation and how would you generalise that to A of any size and the square matrix I of any size?

Just for reference:
(3) prove AI = A and IA = A, where I represents identity matrices of the appropriate (possibly different) sizes
Yeah, that working is essentially correct.









(And thanks for reposting the Q. here – saved me from having to go back and find it. :))
 
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1008

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InteGrand

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Yeah, never heard that before haha. But thanks, it makes sense.

And just for greater conceptual understanding, is there another way to prove it? Any hints?

If I knew about the Kronecker delta, I'd be able to solve so many problems more quickly omg
Yeah, there are other ways of proving it. It's easiest to just think about it intuitively I guess. The i-j entry of AI is the i-th row of A dotted with the j-th column of I. The j-th column of I is just the j-th standard basis vector ej for Rn (or Cn in complex case) and dotting any vector with the j-th standard basis vector just returns the j-th entry of that vector. Hence

ij entry of AI = i-th row of A dotted with j-th column of I
= j-th entry of (i-th row vector of A)
= aij (essentially by definition of aij),

so the ij entry of AI = ij entry of A, so AI = A.
 
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1008

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Yeah, there are other ways of proving it. It's easiest to just think about it intuitively I guess. The i-j entry of AI is the i-th row of A dotted with the j-th column of I. The j-th column of I is just the j-th standard basis vector ej for Rn (or Cn in complex case) and dotting any vector with the j-th standard basis vector just returns the j-th entry of that vector. Hence

ij entry of AI = i-th row of A dotted with j-th column of I
= j-th entry of (i-th row vector of A)
= aij (essentially by definition of aij),

so the ij entry of AI = ij entry of A, so AI = A.
Yeah, that makes sense too. Just a question, how do you go about studying all these topics? Do you have a textbook you prefer, or is it more like you go online to clarify concepts? Or is it a website you go to?
 
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InteGrand

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Yeah, that makes sense too. Just a question, how do you go about studying all these topics? Do you have a textbook you prefer, or is it more like you go online to clarify concepts? Or is it a website you go to?
I guess textbooks, online resources / YouTube videos, and practice are good ways to study these types of things (as well as spending time thinking about them, mentally asking Q's about the topics and trying to answer them or investigating them, etc.).
 
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1008

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I guess textbooks, online resources / YouTube videos, and practice are good ways to study these types of things (as well as spending time thinking about them, mentally asking Q's about the topics and trying to answer them or investigating them, etc.).
Do you have any textbook/resources in mind for vectors and matrices?

And how would you do this?

 
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leehuan

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Do you have any textbook/resources in mind for vectors and matrices?

And how would you do this?

For the first part just evaluate the determinant down the third column by brute force. The determinants of the minors (2x2 matrices) are actually really tidy.

I think InteGrand provided a proof to part b) somewhere.

(Sorry about last night... I ended up going to sleep!!!)
 

1008

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For the first part just evaluate the determinant down the third column by brute force. The determinants of the minors (2x2 matrices) are actually really tidy.

I think InteGrand provided a proof to part b) somewhere.

(Sorry about last night... I ended up going to sleep!!!)
That's alright :) and thanks. I guess the main trick is knowing where to get your determinant from in this question. Btw, do you still have your working for that particular question (proving all those properties of matrices)? Just wanna compare my working to yours...
 

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