# Writing linear maps in matrix formWatch

Announcements
#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
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. , the question is suggesting that you consider the i-th component of the image of this mapping directly, namely . Given what you're told about the linear map in the question, what is the i-th component of ?

You then want to solve that equation for so you'll need to write the LHS in the form [stuff with 's, kronecker deltas etc] .
0
#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. , the question is suggesting that you consider the i-th component of the image of this mapping directly, namely . Given what you're told about the linear map in the question, what is the i-th component of ?

You then want to solve that equation for so you'll need to write the LHS in the form [stuff with , kronecker deltas etc] .
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
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. Same principle though.
0
#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. Same principle though.
Ahh yes thanks for the help.
0
X

new posts
Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

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

### University open days

• University of Bristol
Wed, 23 Oct '19
• University of Exeter
Wed, 23 Oct '19
• University of Nottingham
Wed, 23 Oct '19

### Poll

Join the discussion

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%