Linear transformation using Composite Rule

In summary, a linear transformation is a mathematical process that preserves the structure of inputs while also maintaining certain operations. The Composite Rule is a method for combining linear transformations, commonly used in fields such as computer graphics and engineering. It allows for more efficient calculations and reduces the risk of errors. However, it cannot be applied to non-linear transformations.
  • #1
foreverdream
41
0
I have got

t: P3 → P3
p(x) → p(x) + p(2)

and s: P3 → P2
p(x) → p’(x)

thus
s o t gives P3→ P2gives
p(x) → p’(x)

next part says :

use the composite rule to find a matrix representation of the linear transformation s o t when

t: P3 → P3
p(x) : p(x) + p(2)

and s: P3 →P2
p(x) → p’(x)

now my answer at the back says : it follows from the composite rule that the matrix of s o t with respect to standard bases for the domain and codomain is :

(matrix 1) composite (matrix 2 ) = (matrix 3)
¦0 1 0¦ composite ¦2 2 4 ¦ ¦ 0 1 0¦
¦0 0 2¦ composite ¦0 1 0 ¦= ¦0 0 2¦
¦0 0 1¦

Please explain how? apologies for using ¦ symbol instead of ( )for matrix as I don't know how to
 
Physics news on Phys.org
  • #2
Hey foreverdream.

I'm not exactly sure what your p's and P's are. I'm guessing these are just matrix operators on a vector because a linear function basically has that property.

Are your P's some kind of space? I haven't come across these kinds of terms before. Instead of writing out the math if you know say the wikipedia page or another page (like a university website) that has these terms then you could post a link, or if the questions are online then you could link to the PDF.
 
  • #3
oh sorry - its polynomials of degree 2 and 3

p(x) = has standard basis {1,x,x^2} = a+bx+cx^2 and p'(x) is its derivatives.

Hope I am making sense
 
  • #4
I'm not sure what the "composite rule" is. Does the requirement that you should use the composite rule mean that you should find the matrix representations of s and t separately, and then multiply them?
Fredrik said:
The relationship between linear operators and matrices is explained e.g. in post #3 in this thread. (Ignore the quote and the stuff below it).
By the way, since this is a textbook-style problem, it should be in the homework forum. We are only allowed to give you hints, not complete solutions. You will have to show an attempt to use those hints before we can give you more help. Basically, for this type of questions, you should always show your work up to the point where you get stuck.

If you want to learn how to include nice-looking matrices in your post, see the LaTeX guide.
 
  • #5
oh no- its not a home work question- I am studying pur mathematics and this is one of the explanation - where they straightaway gave matrix- but I am not sure how do you arrive at it. this part of the book covers composite matrix- so like in this one if one is s and another one is t , what would be (sot) or (tos)

the rule says (sot)= (matrix of s) o (matrix of t)
I worked through other simpler examples such as s: 3x+y and t : 2x+3
etc.
I also understand their explanation of the first part but second part is not clear- so hoping someone can simplify this for me.
 
  • #6
i see if I can paste a image of textbook here
 
  • #7
here it is - its in a word document formate hope you can read it. You'll see that there is a problem and answer - which I fail to understand
 
Last edited:
  • #8
Apologies if this still seem like a homework question- I am happy for you to move it- if that's the case
 
  • #9
OK, you need to find the matrix representations of s and t, and then multiply them.
 
  • #10
That's what I am not sure of. How might you do it?
 
  • #11
See the post I linked to in post #4.
 
  • #12
Ok - thanks.
 
  • #13
Number 4 post is about orthogonal vectors? Is that the one you want me to look at?
 
  • #14
Number 4 post is about orthogonal vectors? Is that the one you want me to look at?
 
  • #15
foreverdream said:
Number 4 post is about orthogonal vectors? Is that the one you want me to look at?
I don't know what you're looking at. None of my posts in this thread mentions orthogonality. Post #4 tells you where you can find an explanation of the relationship between linear operators and matrices.
 

FAQ: Linear transformation using Composite Rule

1. What is a linear transformation?

A linear transformation is a mathematical process that transforms a set of inputs (usually represented as vectors) into a set of outputs. It has the property of preserving the basic structure of the input, such as the shape and relative position of points, while also maintaining certain operations, such as addition and scalar multiplication.

2. What is the Composite Rule for linear transformations?

The Composite Rule for linear transformations is a method for combining two or more linear transformations into a single transformation. It involves applying the first transformation to the input, and then applying the second transformation to the output of the first transformation. This results in a new transformation that is equivalent to performing both transformations separately.

3. How is the Composite Rule used in real-world applications?

The Composite Rule is commonly used in areas such as computer graphics, engineering, and physics, where multiple linear transformations are often required to achieve a desired result. For example, in computer graphics, the Composite Rule can be used to transform an object's position, rotation, and scale simultaneously.

4. What are the benefits of using the Composite Rule for linear transformations?

The Composite Rule allows for complex transformations to be broken down into simpler steps, making it easier to understand and implement. It also allows for more efficient calculations and reduces the risk of errors, as each transformation can be tested and verified separately before being combined.

5. Can the Composite Rule be applied to non-linear transformations?

No, the Composite Rule is specifically designed for linear transformations, which have the property of preserving the basic structure of the input. Non-linear transformations do not have this property, so the Composite Rule cannot be applied to them.

Similar threads

Replies
10
Views
2K
Replies
3
Views
829
Replies
3
Views
2K
Replies
25
Views
4K
Back
Top