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Math Help - Basic linear transformation help

  1. #1
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    Basic linear transformation help

    Given V_R = R^2, then let V_R have the standard basis B=B'={(1,0), (0,1)} and let L_theta((1,0)) = (cos theta, sin theta) , L_theta((0,1)) = (-sin theta, cos theta).

    I understand how the bases work, but how would you go about obtaining the new vector by way of this linear transformation? For example, how you would go about computing L_(pi/4) (2,1)?

    I'm thinking you would need to multiply the first L_theta by two since the first coordinate in the input is 2, but how do you "combine" L_theta((1,0)) and L_theta ((0,1)) to obtain your final answer?

    Or, in lieu of giving an answer or hint, can someone at least tell me what I can search under to find more information on this? I'm not sure what to call it, and Googling "linear transformations" only gives theoretical stuff. I haven't been able to find anything like this.
    Last edited by scosgurl; April 25th 2009 at 09:37 AM.
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  2. #2
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    Quote Originally Posted by scosgurl View Post
    Given V_R = R^2, then let V_R have the standard basis B=B'={(1,0), (0,1)} and let L_theta((1,0)) = (cos theta, sin theta) , L_theta((0,1)) = (-sin theta, cos theta).

    I understand how the bases work, but how would you go about obtaining the new vector by way of this linear transformation? For example, how you would go about computing L_(pi/4) (2,1)?
    It is, \begin{bmatrix} \cos \tfrac{\pi}{4} & -\sin \tfrac{\pi}{4} \\ \sin \tfrac{\pi}{4} & \cos \tfrac{\pi}{4} \end{bmatrix} \begin{bmatrix} 2\\1 \end{bmatrix}
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