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Math Help - Basic Linear Algebra - Linear Transformations Help

  1. #1
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    Basic Linear Algebra - Linear Transformations Help

    So I know that a transformation is linear if T(u+v) = T(u) + T(v) and T(kv) = kT(v).
    I'm given a system
    y1 = b - c
    y2 = ac
    y3 = a - b
    and to be honest I have no idea where to start.
    Any help is appreciated.
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  2. #2
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    Are you trying to determine if \displaystyle y_1, \ y_2, \ \mbox{and} \ y_3 are each a linear transformation?
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  3. #3
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    No, I think it's referring to the system as a whole. Maybe interpreted as a matrix? I'm not quite sure. My book is somewhat vague on this topic so I really quite confused.

    EDIT: The answer in the back of the book says non-linear, so yes, I'd assume it's referring if the whole taken as a transformation is linear.
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  4. #4
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    \displaystyle \begin{bmatrix}<br />
 b-c\\ <br />
 ac\\ <br />
a-b <br />
\end{bmatrix},\ \begin{bmatrix}<br />
 y-z\\ <br />
 xz& \\ <br />
x-y & <br />
\end{bmatrix}\in S

    \displaystyle \begin{bmatrix}<br />
 b-c\\ <br />
 ac\\ <br />
a-b <br />
\end{bmatrix} +\begin{bmatrix}<br />
 y-z\\ <br />
 xz& \\ <br />
x-y & <br />
\end{bmatrix}=\begin{bmatrix}<br />
b-c +y-z\\ <br />
 ac+xz& \\ <br />
a-b+x-y & <br />
\end{bmatrix}=\begin{bmatrix}<br />
 (b+y)-(c+z)\\ <br />
 ac+xz& \\ <br />
(a+x)-(b+y) & <br />
\end{bmatrix}\neq \begin{bmatrix}<br />
 (b+y)-(c+z)\\ <br />
 (a+x)(c+z)& \\ <br />
(a+x)-(b+y) & <br />
\end{bmatrix}
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  5. #5
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    Thank you very much!
    For some reason it never occurred to me to just pick another set of points to prove/disprove it.
    Now, I also have the system
    y1 = 2b, y2 = 3c, y1 = a ,
    Am I correct in saying that this is linear b/c
    2b + 2y = 2(b+y), 3(c + z) = 3c + 3z, and so forth?

    Also, considering the system
    y1 = 2x, y2 = x + 2, y3 = 2x, I'm confused again...
    y1 = y3 clearly are linear, and I'm a little confused about proving how y2 is/isn't.
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  6. #6
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    Quote Originally Posted by Lord Voldemort View Post
    Thank you very much!
    For some reason it never occurred to me to just pick another set of points to prove/disprove it.
    Now, I also have the system
    y1 = 2b, y2 = 3c, y1 = a ,
    Am I correct in saying that this is linear b/c
    2b + 2y = 2(b+y), 3(c + z) = 3c + 3z, and so forth?

    Also, considering the system
    y1 = 2x, y2 = x + 2, y3 = 2x, I'm confused again...
    y1 = y3 clearly are linear, and I'm a little confused about proving how y2 is/isn't.
    Yeah correct for the first one. You also need to show scalar multiplication.

    \displaystyle (x+2)+(\beta+2)=x+\beta+4\neq (x+\beta)+2

    Equations of the form y = mx + b are affine transformations.
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  7. #7
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    I'm not quite sure what an affine transformation is yet, is it just a nonlinear one?
    (In regards to your last explanation)
    Basically, y2 adds two to the the value input, but because adding two to both values and then adding the values as compared to adding the values as one value in and of itself and then adding two to that results in an inequality, it's not linear?
    I guess I didn't really know how to approach the issue.

    Thanks so much, I was a bit unclear on this but I definitely get this now!
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