Results 1 to 3 of 3

Math Help - Linear transformation problem

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    11

    Linear transformation problem

    A linear transformation is given: T:R^3 -> R^3 such as:
    T(1,1,1)=(1,1,1)
    T(0,1,0)=(0,1,0)
    T(1,0,2)=(1,0,1)
    Find T(x,y,z) and [T]_E (E={(1,0,0),(0,1,0),(0,0,1)})
    I know that first you must prove that (1,1,1),(0,1,0) and(1,0,2) are linearly independent which is not a problem my problem is finding T(x,y,z)
    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,306
    Thanks
    1283
    Quote Originally Posted by wicked View Post
    A linear transformation is given: T:R^3 -> R^3 such as:
    T(1,1,1)=(1,1,1)
    T(0,1,0)=(0,1,0)
    T(1,0,2)=(1,0,1)
    Find T(x,y,z) and [T]_E (E={(1,0,0),(0,1,0),(0,0,1)})
    I know that first you must prove that (1,1,1),(0,1,0) and(1,0,2) are linearly independent which is not a problem my problem is finding T(x,y,z)
    Thanks in advance.
    Since (1,1,1), (0,1,0), and (1,0,2) are linearly independent (since you say "not a problem" I presume you have already proved that) and there are three vectors, they form a basis for R^3. In particular, that means that any vector (x,y,z)= a(1,1,1)+ b(0,1,0)+ c(1,0,2) where a, b, and c are numerical functions of x, y, and z. After you have found those numbers, T(x,y,z)= aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2010
    Posts
    11
    Quote Originally Posted by HallsofIvy View Post
    (x,y,z)= a(1,1,1)+ b(0,1,0)+ c(1,0,2) where a, b, and c are numerical functions of x, y, and z. After you have found those numbers, T(x,y,z)= aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1).
    Okay,so this is what I get
    a+c=x     =>     a=2x-z
    a+b=y      =>     b=y+z-2x
    a+2c=z   =>     c=z-x

    aT(1,1,1)+ bT(0,1,0)+ cT(1,0,2)= a(1,1,1)+ b(0,1,0)+ c(1,0,1) =
    (2x-z)(1,1,1)+(y+z-2x)(0,1,0)+(z-x)(1,0,1)=(2x-z,2x-z,2x-z)+ (0,y+z-2x,0)+ (z-x,0,z-x)=(x,y,x) right?

    So the [T]_E matrix is :

    ( [T(1,0,0)]_E [T(0,1,0)]_E [T(0,0,1)]_E) =

    1 0 0
    0 1 0 (this is the [T]^E_E matrix(=[T]_E))
    1 0 0
    Is it correct?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. linear transformation problem
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 1st 2010, 03:13 AM
  2. Linear Transformation problem
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: February 6th 2010, 04:58 AM
  3. Linear transformation problem
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 17th 2008, 01:09 PM
  4. Linear Transformation problem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 3rd 2008, 07:20 PM
  5. Linear Transformation Problem...
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 20th 2007, 08:48 PM

Search Tags


/mathhelpforum @mathhelpforum