Results 1 to 4 of 4

Thread: Linear transformation between function spaces

  1. #1
    tun
    tun is offline
    Newbie
    Joined
    Aug 2009
    Posts
    2

    Linear transformation between function spaces

    Hi all.

    I am a beginner to real maths and am looking at a problem from a book by Loomis and Sternberg.

    Let $\displaystyle T$ be the linear transformation from $\displaystyle \mathbb{R}^B$ to $\displaystyle \mathbb{R}^A$ such that $\displaystyle T(f) = f \circ \varphi $.

    I need to show that $\displaystyle T$ is an isomorphism if $\displaystyle \varphi$ is a bijection by showing that:

    (a) $\displaystyle \varphi$ injective -> $\displaystyle T$ surjective;

    (b) $\displaystyle \varphi$ surjective -> $\displaystyle T$ injective.

    I'm not really sure where to start, and I can't even see the link between the properties of the 2 mappings; any hints would be greatly appreciated.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2008
    Posts
    81
    Is $\displaystyle \mathbb{R}^A=\{f\text{ such that }f: A \rightarrow \mathbb{R}\}$ (I hate this notation), and in what sense do you mean isomorphism?

    If I am right on the notation, I suppose it must be that $\displaystyle \phi: A \rightarrow B$. Suppose $\displaystyle \varphi$ is injective. Let $\displaystyle g:A\rightarrow \mathbb{R}$ be any function.
    We need to find a function $\displaystyle f: B\rightarrow \mathbb{R}$ so that $\displaystyle g=f\circ \varphi$.
    To do this, first note that since $\displaystyle \varphi$ is injective, it is actually a bijection $\displaystyle A\rightarrow \text{Range}(\varphi)$,
    so there exists $\displaystyle \gamma:\text{Range}(\varphi)\rightarrow A$ such that $\displaystyle \gamma \circ \varphi(a)=a$ for all $\displaystyle a\in A$.
    Then define $\displaystyle f(b)=0$ if $\displaystyle b\notin \text{Range}(\varphi)$ and $\displaystyle f(b)=g(\gamma(b))$ otherwise. Then $\displaystyle f \circ \phi (a)=f(\phi(a))=g(\gamma(\phi(a)))=g(a)$. Since we have found a preimage for an arbitrary element, the function is onto.

    See if you can do (b) on your own now.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    tun
    tun is offline
    Newbie
    Joined
    Aug 2009
    Posts
    2
    Thanks for that, I was on completely the wrong lines - I was thinking it was something to do with the linearity. You were right about the notation. Out of interest, what do you use to represent spaces like $\displaystyle \mathbb{R}^A$?

    As for (b), if I'm right then it's easier than (a):

    Let $\displaystyle f$ and $\displaystyle g$ be functions from $\displaystyle B$ to $\displaystyle \mathbb{R}$. We need to show that $\displaystyle T(f) = T(g) \Rightarrow f = g$, then T is injective.

    $\displaystyle T(f) = T(g)$ means that $\displaystyle f \circ \varphi = g \circ \varphi$, so that f and g are equal for all elements in $\displaystyle B$ that are in the range of $\displaystyle \varphi$. If $\displaystyle \varphi$ is surjective, then the range of $\displaystyle \varphi$ is all of $\displaystyle B$ and so $\displaystyle f$ and $\displaystyle g$ are equal everywhere in their domain.

    Does this seem right? Thanks for your help!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jul 2008
    Posts
    81
    That looks right to me!

    As for the notation, I just define a set to be all such functions, i.e. let $\displaystyle \mathcal{F}=\{ f:A\rightarrow B\text{ such that } f\text{ satisfies...}\}$.
    It may be less "slick" but it sure prevents any and all confusion about what you are saying.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Aug 1st 2011, 10:00 PM
  2. Transformation between two non-aligned 2-dimensional spaces
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: Nov 25th 2010, 08:30 AM
  3. linear algebra Gram-Schmidt in Function Spaces problem
    Posted in the Advanced Algebra Forum
    Replies: 58
    Last Post: Aug 20th 2010, 10:18 AM
  4. Sum of Linear Spaces
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Jun 18th 2010, 09:11 AM
  5. Linear Spaces
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Oct 10th 2009, 09:21 AM

Search Tags


/mathhelpforum @mathhelpforum