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Math Help - Linear Transformations

  1. #1
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    Linear Transformations

    Hello everyone,

    I am currently having trouble with this problem. Any help would be greatly appreciated!

    (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.

    T is the counterclockwise rotation of 45 degrees in R^2, v = (2,2)
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  2. #2
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    Quote Originally Posted by larson View Post
    Hello everyone,

    I am currently having trouble with this problem. Any help would be greatly appreciated!

    (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image.

    T is the counterclockwise rotation of 45 degrees in R^2, v = (2,2)
    Notice that T is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let \bold{i} = (1,0) \text{ and }\bold{j} = (0,1) (just remember to think of them as coloumns, when we deal with \mathbb{R}^n we sometimes things of vectors as an n\times 1 matrix). Compute T(\bold{i}) and T(\bold{j}) (again as coloumns) and form the 2\times 2 matrix [ T(\bold{i}) ~ ~ ~ T(\bold{j})] .
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Notice that T is a linear transformation. You can get the matrix of the transofrmation by figuring out how it acts on its standard basis. Let \bold{i} = (1,0) \text{ and }\bold{j} = (0,1) (just remember to think of them as coloumns, when we deal with \mathbb{R}^n we sometimes things of vectors as an n\times 1 matrix). Compute T(\bold{i}) and T(\bold{j}) (again as coloumns) and form the 2\times 2 matrix [ T(\bold{i}) ~ ~ ~ T(\bold{j})] .
    I'm sorry... I'm still kind of confused. How do I find T?
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  4. #4
    Senior Member Twig's Avatar
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    hi

    hi


     \left[\begin{matrix} 1 \\ 0 \end{matrix}\right] rotates into  \left[\begin{matrix} cos(\theta) \\ sin(\theta) \end{matrix}\right] and  \left[\begin{matrix} 0 \\ 1 \end{matrix}\right] rotates into  \left[\begin{matrix} -sin(\theta) \\ cos(\theta) \end{matrix}\right]




    Rotation matrix here will be:
    A =  \left[ \begin{matrix} cos(\frac{\pi}{4}) & -sin(\frac{\pi}{4}) \\ sin(\frac{\pi}{4}) & cos(\frac{\pi}{4}) \end{matrix} \right]

    Since  A \left[ \begin{matrix} 1 \\ 0 \end{matrix}\right] = \left[\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right] and  A \left[ \begin{matrix} 0 \\ 1 \end{matrix}\right] = \left[\begin{matrix} -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}\right]

    Now calculate  Av
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