Peter8837
Badges: 1
Rep:
?
#1
Report Thread starter 6 years ago
#1
Hi guys,

This is something that has confused me - the problem is, we're just not taught explicitly how to do this in lectures. Take questions 8 and 9 on http://www.damtp.cam.ac.uk/user/examples/A1b.pdf as an example.

How do you write those in matrix form?? What's the way to do these?

Any help is appreciated.

Thanks.
0
reply
Farhan.Hanif93
Badges: 18
Rep:
?
#2
Report 6 years ago
#2
(Original post by Peter8837)
Hi guys,

This is something that has confused me - the problem is, we're just not taught explicitly how to do this in lectures. Take questions 8 and 9 on http://www.damtp.cam.ac.uk/user/examples/A1b.pdf as an example.

How do you write those in matrix form?? What's the way to do these?

Any help is appreciated.

Thanks.
You're simply looking for an expression for the ij-th entry of the matrix S (in Q8). Since S is a matrix s.t. \mathbf{x} \mapsto \mathbf{x'}=S\mathbf{x} , the question is suggesting that you consider the i-th component of the image of this mapping directly, namely x'_i = [S\mathbf{x}]_i = S_{ij}x_j. Given what you're told about the linear map in the question, what is the i-th component of \mathbf{x'}?

You then want to solve that equation for S_{ij} so you'll need to write the LHS in the form [stuff with x_i's, kronecker deltas etc] \times x_j.
0
reply
Peter8837
Badges: 1
Rep:
?
#3
Report Thread starter 6 years ago
#3
(Original post by Farhan.Hanif93)
You're simply looking for an expression for the ij-th entry of the matrix S (in Q8). Since S is a matrix s.t. \mathbf{x} \mapsto \mathbf{x'}=S\mathbf{x} , the question is suggesting that you consider the i-th component of the image of this mapping directly, namely x'_i = [S\mathbf{x}]_i = S_{ij}x_j. Given what you're told about the linear map in the question, what is the i-th component of \mathbf{x'}?

You then want to solve that equation for S_{ij} so you'll need to write the LHS in the form [stuff with x_i's, kronecker deltas etc] \times x_j.
Ok so x'_i = (delta_ij + (lambda).a_i.b_j)x_j

=> S_ij = delta_ij + (lambda).a_i.b_j.

Is that right? Now how do we write S_ij in matrix form? Firstly, is it a 2x2 matrix or 3x3 as the question asks for S(lambda, a, b)...?

Do we consider each entry separately? So:

S_11 = 1 + (lambda).a_1.b_1.
.
.
etc

Is that the right approach?

---

Note, following this through and finding det S (which is quite long but most things cancel out) I get:

det (S) = 1 + lambda (a_1.b_1 + a_2.b_2 + a_3.b_3) = 1 + lambda (a.b)

Is that right?

EDIT: That is right since det (S) = 1 as a and b are orthogonal, so the map is indeed area preserving.
0
reply
Farhan.Hanif93
Badges: 18
Rep:
?
#4
Report 6 years ago
#4
(Original post by Peter8837)
Ok so x'_i = (delta_ij + (lambda).a_i.b_j)x_j

=> S_ij = delta_ij + (lambda).a_i.b_j.

Is that right? Now how do we write S_ij in matrix form? Firstly, is it a 2x2 matrix or 3x3 as the question asks for S(lambda, a, b)...?

Do we consider each entry separately? So:

S_11 = 1 + (lambda).a_1.b_1.
.
.
etc

Is that the right approach?
Yep. You're told that S represent a map from R2 to R2, so it'll be a 2x2 matrix.

Note, following this through and finding det S (which is quite long but most things cancel out) I get:

det (S) = 1 + lambda (a_1.b_1 + a_2.b_2 + a_3.b_3) = 1 + lambda (a.b)

Is that right?

EDIT: That is right since det (S) = 1 as a and b are orthogonal, so the map is indeed area preserving.
It's a 2x2 matrix so there shouldn't be any "_3" terms, and this reduces the calculation a fair bit. :p: Same principle though.
0
reply
Peter8837
Badges: 1
Rep:
?
#5
Report Thread starter 6 years ago
#5
(Original post by Farhan.Hanif93)
Yep. You're told that S represent a map from R2 to R2, so it'll be a 2x2 matrix.

...

It's a 2x2 matrix so there shouldn't be any "_3" terms, and this reduces the calculation a fair bit. :p: Same principle though.
Ahh yes thanks for the help.
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • University of Bristol
    Undergraduate Open Afternoon Undergraduate
    Wed, 23 Oct '19
  • University of Exeter
    Undergraduate Open Day - Penryn Campus Undergraduate
    Wed, 23 Oct '19
  • University of Nottingham
    Mini Open Day Undergraduate
    Wed, 23 Oct '19

Have you made up your mind on your five uni choices?

Yes I know where I'm applying (152)
59.61%
No I haven't decided yet (58)
22.75%
Yes but I might change my mind (45)
17.65%

Watched Threads

View All