Linear transformation using Composite Rule

[foreverdream]

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

 

[chiro]

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.

 

[foreverdream]

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

 

[Fredrik]

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?

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.

 

[foreverdream]

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.

 

[foreverdream]

i see if I can paste a image of textbook here

 

[foreverdream]

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

 

[foreverdream]

Apologies if this still seem like a homework question- I am happy for you to move it- if that's the case

 

[Fredrik]

OK, you need to find the matrix representations of s and t, and then multiply them.

 

[foreverdream]

That's what I am not sure of. How might you do it?

 

[Fredrik]

See the post I linked to in post #4.

 

[foreverdream]

Ok - thanks.

 

[foreverdream]

Number 4 post is about orthogonal vectors? Is that the one you want me to look at?

 

[foreverdream]

Number 4 post is about orthogonal vectors? Is that the one you want me to look at?

 

[Fredrik]

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.