Results 1 to 3 of 3

Thread: Linear Transformation

  1. #1
    Junior Member
    Nov 2008

    Linear Transformation

    If is a linear transformation such that
    1 2 11 14 6 and 4 -1 17 2 6 ,
    what is the standard matrix of T?

    Any help is appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Sep 2009
    I dont know what standard matrix is, but I am going to assume it means the matrix of
    T with respect to the standard basis.

    Now what you need to find is what T does to the standard basis

    $\displaystyle T\left(\begin{bmatrix}1\\2\end{bmatrix}\right)=T\l eft(\begin{bmatrix}1\\0\end{bmatrix}+2\cdot\begin{ bmatrix}0\\1\end{bmatrix}\right)=T\left(\begin{bma trix}1\\0\end{bmatrix}\right)+2T\left(\begin{bmatr ix}0\\1\end{bmatrix}\right)=\begin{bmatrix}11\\14\ \6\end{bmatrix}$

    And :

    $\displaystyle T\left(\begin{bmatrix}4\\-1\end{bmatrix}\right)=T\left(4\cdot\begin{bmatrix} 1\\0\end{bmatrix}-1\cdot\begin{bmatrix}0\\1\end{bmatrix}\right)=4T\l eft(\begin{bmatrix}1\\0\end{bmatrix}\right)-1T\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\b egin{bmatrix}17\\2\\6\end{bmatrix}$

    Multiplying the lower formula by 2 and adding to the upper one gives:

    $\displaystyle 9T\left(\begin{bmatrix}1\\0\end{bmatrix}\right)+0= \begin{bmatrix}11\\14\\6\end{bmatrix}+2\cdot\begin {bmatrix}17\\2\\6\end{bmatrix}$

    So that
    $\displaystyle T\left(\begin{bmatrix}1\\0\end{bmatrix}\right)=\fr ac{1}{9}\cdot\begin{bmatrix}54\\18\\18\end{bmatrix }$

    Now find $\displaystyle T\left(\begin{bmatrix}0\\1\end{bmatrix}\right)$

    And the standard matrix would then be:

    $\displaystyle \begin{bmatrix}T\left(\begin{bmatrix}1\\0\end{bmat rix}\right) & T\left(\begin{bmatrix}0\\1\end{bmatrix}\right)\end {bmatrix}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Jan 2009
    As above, I repeat equations.

    $\displaystyle T(\vec{x})=A\vec{x},A=[T(\vec{e}_1)\ \ T(\vec{e}_2)]$

    $\displaystyle T(\vec{e}_1)=\frac{1}{9}\cdot\begin{bmatrix} *45* \\18\\18\end{bmatrix}=\begin{bmatrix} 5 \\2\\2\end{bmatrix}\ ,\ T(\vec{e}_2)=\begin{bmatrix} 3 \\6\\2\end{bmatrix}\ ,\ A=\begin{bmatrix} 5&3\\2&6\\2&2\end{bmatrix}$
    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. Example of a linear transformation T:R^4 --> R^3?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 5th 2011, 07:04 PM
  3. linear transformation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 28th 2009, 06:40 AM
  4. Linear Algebra.Linear Transformation.Help
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Mar 5th 2009, 01:14 PM
  5. Linear Transformation
    Posted in the Pre-Calculus Forum
    Replies: 10
    Last Post: May 25th 2008, 12:14 AM

Search Tags

/mathhelpforum @mathhelpforum