Results 1 to 11 of 11

Math Help - Operator

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1

    Operator

    Need some help with this... doesn't seem to be a decent example in my book. Appreciate any help!
    Attached Thumbnails Attached Thumbnails Operator-555.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by gralla55 View Post
    Need some help with this... doesn't seem to be a decent example in my book. Appreciate any help!

    1) Apply the transformation T to each and every element of E and write the result as a linear combination of E itself;

    2) Take the transpose , if you apply transformations from the left and vectors from the right), or directly (otherwise) the coefficients matrix of the above: this is [T]_E.

    For example:
    T(1)=1+1 = 2=2\cdot 1+0\cdot x+0\cdot x^2+0\cdot x^3

    T(x)=x+x-2=-2+2x=(-2)\cdot 1+2\cdot x +0\cdot x^2+0\cdot x^3 ...

    So the first two columns of [T]_E are \begin{pmatrix}2\\0\\0\\0\end{pmatrix}\,,\,\,\begi  n{pmatrix}\!\!-2\\2\\0\\0\end{pmatrix} (or the first two rows if you write the map on the right of the vector).

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    Thanks for your reply! But how do you get:

    T(x) = -2 + 2x ? Shouldn't that just equal 2?

    Thanks again.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Quote Originally Posted by gralla55 View Post
    Thanks for your reply! But how do you get:

    T(x) = -2 + 2x ? Shouldn't that just equal 2?

    Thanks again.
    Yes, that must have been a typo. Since T(f)= f(x)+ f(2- x), T(x)= x+ (2- x)= 2.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    In that case 2 is the answer for all four. And a linear combination would just be every element of E times 1/2 times each column vector? And won't the transpose of this matrix is the same as the matrix itself?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by gralla55 View Post
    In that case 2 is the answer for all four.


    Why? T(x^2)=x^2+(2-x)^2=2x^2-4x+4 ...

    Tonio


    And a linear combination would just be every element of E times 1/2 times each column vector? And won't the transpose of this matrix is the same as the matrix itself?

    .
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    Don't you just substitute the "x" for "x^2" ? I don't see why the ^2 suddenly goes outside the parantheses. But of course, I might be wrong here...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    What confuses me is the notation T(f(x))... In case you're right:

    T(1) = 1 + (2-1) = 2
    T(x) = x + (2x-2) = 2
    T(x^2) = 4 - 4x + 2x^2
    T(x^3) = 8 - 12x + 8x^2 - 2x^3

    How to proceed?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by gralla55 View Post
    What confuses me is the notation T(f(x))... In case you're right:

    T(1) = 1 + (2-1) = 2
    T(x) = x + (2x-2) = 2
    T(x^2) = 4 - 4x + 2x^2
    T(x^3) = 8 - 12x + 8x^2 - 2x^3

    How to proceed?

    Write each outcome as a linear combination of the given basis and thus the coefficients in each case become the wanted matrix's columns...

    Tonio
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    So:

    2 2 4 8
    0 0 -4 -12
    0 0 2 8
    0 0 0 -2

    and then I take the transpose:

    2 0 0 0
    2 0 0 0
    4 -4 2 0
    8 -12 8 -2

    And that's it?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Oct 2009
    Posts
    195
    Thanks
    1
    Nevermind, I figured it out! And I also did a mistake, the correct answer is:

    2 2 4 8
    0 0 -4 -12
    0 0 2 6
    0 0 0 0

    Now, how can I find a solution to T(f) = (x-1)^2 ?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Zero Operator
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 4th 2011, 12:23 AM
  2. Operator in R^n
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 21st 2010, 08:22 AM
  3. D Operator
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: September 15th 2009, 11:54 AM
  4. Operator Help
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: June 25th 2009, 07:13 PM
  5. The use of an operator
    Posted in the Algebra Forum
    Replies: 5
    Last Post: March 20th 2008, 01:47 PM

Search Tags


/mathhelpforum @mathhelpforum