Maths 3A/3B

[anotherworld2b]

I'll ask about q21 if i have any more queries about it.
I was wondering for q15 am I doing something wrong? Because im getting the wrong answers for  x and y so far. Also for q17 how do I prove generally?

Where are you up to with Q21? Have you started by multiplying the vector I suggested above with the matrix given? That will apply the linear transformation, did you get that or is that operation troubling you?

Question 15 isn't really attackable with the method you used, you are better off using simultaneous!



Using the statement given, we conclude the following:



Do a similar thing for the multiplication of BA (the other way around), then you'll have four sets of simultaneous equations. Solve each to obtain an answer

For Q16 ,you have the right idea, but remember that matrix multiplication is not commutative! That is:



When you factored, you put the P out the front, it should have been out the back to preserve the initial order of the matrices, try again with:



Question 21 is asking you to consider general vectors and do a general proof of the statements given, kind of like a standard algebraic proof! so, consider A and B as non-singular (invertible) square matrices where \(AB=BA\), and prove generally that:



Let me know how you go!

[jamonwindeyer]

I'll ask about q21 if i have any more queries about it.
I was wondering for q15 am I doing something wrong? Because im getting the wrong answers for  x and y so far. Also for q17 how do I prove generally?

Hmm, I don't see any errors in your working immediately, what are the answers supposed to be?

An example for 17a) to show you a general proof:



At no stage do I specify what A or B should be, I've proved it generally!

[anotherworld2b]

The answers for q15 are:
X= -1 , y= -2 , p= -5 , q= 7 , r= -7 , s=2 but now sure why.
Oh okay I get how to do q17 now thank you 
I also want to ask for q13 what would be the best way to do it?
I also tried q14 but i also got the answer wrong for some reason

 

Hmm, I don't see any errors in your working immediately, what are the answers supposed to be?

An example for 17a) to show you a general proof:



At no stage do I specify what A or B should be, I've proved it generally!

[anotherworld2b]

I was also wondering if i can get help with these two questions. Particularly q13

[jamonwindeyer]

The answers for q15 are:
X= -1 , y= -2 , p= -5 , q= 7 , r= -7 , s=2 but now sure why.
Oh okay I get how to do q17 now thank you 
I also want to ask for q13 what would be the best way to do it?
I also tried q14 but i also got the answer wrong for some reason

Oh! The matrix I gave you in my original explanation had a sign error for Q15, dictation error, sorry! Go back and look at your first line of working, do the matrix multiplication again, fix the sign errors and repeat, your method is 100% correct

Question 13, best way to do it would (I think) just be considering a general matrix \(\begin{bmatrix}a & b\\c &d\end{bmatrix}\) and setting up simultaneous equations just like above!

Question 14, your attempt isn't quite correct. To make it clearer, try drawing a little diagram!

Triangle 1 ------ (Matrix A) -------> Triangle 2
Triangle 1 ------ (Matrix B) -------> Triangle 3

The idea here is to go from Triangle 2 to Triangle 3. We can do this by going back to Triangle 1, then to triangle 3, the matrix therefore being:



where A is the first matrix given, B is the second matrix given

Have you seen this method before? If not happy to explain!

I was also wondering if i can get help with these two questions. Particularly q13

Question 12 should be fairly straightforward, calculate the inverse:



Then multiply it by itself, form 4 equations and then solve for the missing variables! Show me your working if that doesn't quite work!

Then Q13 I addressed above

Hope this helps!! 

[anotherworld2b]

I was able to get q11 right but i could get q13 right. For q14 i followed an example from our textbook but i atill didnt get the answer.
Thank you for your help I rewlly appreciate it.

Oh! The matrix I gave you in my original explanation had a sign error for Q15, dictation error, sorry! Go back and look at your first line of working, do the matrix multiplication again, fix the sign errors and repeat, your method is 100% correct

Question 13, best way to do it would (I think) just be considering a general matrix \(\begin{bmatrix}a & b\\c &d\end{bmatrix}\) and setting up simultaneous equations just like above!

Question 14, your attempt isn't quite correct. To make it clearer, try drawing a little diagram!

Triangle 1 ------ (Matrix A) -------> Triangle 2
Triangle 1 ------ (Matrix B) -------> Triangle 3

The idea here is to go from Triangle 2 to Triangle 3. We can do this by going back to Triangle 1, then to triangle 3, the matrix therefore being:



where A is the first matrix given, B is the second matrix given

Have you seen this method before? If not happy to explain!

Question 12 should be fairly straightforward, calculate the inverse:



Then multiply it by itself, form 4 equations and then solve for the missing variables! Show me your working if that doesn't quite work!

Then Q13 I addressed above

Hope this helps!! 

[jamonwindeyer]

I was able to get q11 right but i could get q13 right. For q14 i followed an example from our textbook but i atill didnt get the answer.
Thank you for your help I rewlly appreciate it.

Q13A should be:



That's if you wanted to do it with two separate equations instead of one big one. Note that the second vector must be a column vector, otherwise the transformation doesn't make sense. Of course what we notice is that the answer is actually just formed with the columns of the vectors in the question, that's a consequence of transforming the standard basis vectors in \(\mathbb{R}^2\), but it's cool if you don't spot that

Have another go

As I said for 14, incorrect working, your worked example in the textbook is slightly different to your question. It's not the triangle \(P'Q'R'\) being transformed to \(P''Q''R''\), it is \(PQR\).



You need the method I showed you above (or something similar)

[anotherworld2b]

oh thank you
I'm not sure how to do q14? 

Q13A should be:



That's if you wanted to do it with two separate equations instead of one big one. Note that the second vector must be a column vector, otherwise the transformation doesn't make sense. Of course what we notice is that the answer is actually just formed with the columns of the vectors in the question, that's a consequence of transforming the standard basis vectors in \(\mathbb{R}^2\), but it's cool if you don't spot that

Have another go

As I said for 14, incorrect working, your worked example in the textbook is slightly different to your question. It's not the triangle \(P'Q'R'\) being transformed to \(P''Q''R''\), it is \(PQR\).



You need the method I showed you above (or something similar)

[jamonwindeyer]

Question 14, your attempt isn't quite correct. To make it clearer, try drawing a little diagram!

Triangle 1 ------ (Matrix A) -------> Triangle 2
Triangle 1 ------ (Matrix B) -------> Triangle 3

The idea here is to go from Triangle 2 to Triangle 3. We can do this by going back to Triangle 1, then to triangle 3, the matrix therefore being:



where A is the first matrix given, B is the second matrix given

This was the explanation I provided earlier. Again, the idea is that to go from Triangle 2 (P'Q'R') to Triangle 3 (P''Q''R''), we need to go back to Triangle 1 by multiplying by the inverse of the matrix given, THEN multiply (on the left) to get to Triangle 3. The product of those matrices must, by definition, be the same as the matrix we need, which is where that expression above comes from

[anotherworld2b]

Then would you do the same thing for Q13b?

This was the explanation I provided earlier. Again, the idea is that to go from Triangle 2 (P'Q'R') to Triangle 3 (P''Q''R''), we need to go back to Triangle 1 by multiplying by the inverse of the matrix given, THEN multiply (on the left) to get to Triangle 3. The product of those matrices must, by definition, be the same as the matrix we need, which is where that expression above comes from

[jamonwindeyer]

Then would you do the same thing for Q13b?

Nope, 13B is the same as 13A

[anotherworld2b]

Thank you very much for your help i finally understand q13. i was wondering for q14 if there was 4 triangles would you have the first matrix that is given being post multipled by the inverse of the 3rd matrix?

I tried q8 and got k=3 or k=-4
But the answer only has k=3?

I also tried q10 but i am not sure where to go

I have a test on matrices tomorrow and i was wondering what would you recommend I should memorise and what should be put on notes? Did you get matrix proofs?

Nope, 13B is the same as 13A

[anotherworld2b]

Questions for ^

[RuiAce]

Questions for ^

[jamonwindeyer]

Thank you very much for your help i finally understand q13. i was wondering for q14 if there was 4 triangles would you have the first matrix that is given being post multipled by the inverse of the 3rd matrix?

I tried q8 and got k=3 or k=-4
But the answer only has k=3?

I also tried q10 but i am not sure where to go

I have a test on matrices tomorrow and i was wondering what would you recommend I should memorise and what should be put on notes? Did you get matrix proofs?

Hmm, not quite sure what you mean there, I'd have to see the question!

Question 8 dictates \(p>0\), which will exclude \(k=-4\) from your possibilities

Good luck with your test tomorrow! I never did matrices at high school level, but my first year algebra course did have a bunch of matrix proofs in the final just relax and do the best you can, sure you'll smash it!