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Question about Vector Planes

  1. Apr 16, 2014 #1
    Hi,

    I am trying to follow an introductory problem in my book for which no solutions are provided and have got stuck. I was wondering whether anyone could tell me how to go about this problem and where I am going wrong.

    The problem starts:

    Consider the eqquations:
    [tex]y_1= x_1+2x_2[/tex]
    [tex]y_2=3x_2[/tex]

    We can view these equations as describing a transformation of the vector x = [itex]\begin{bmatrix}x_1\\x_2\end{bmatrix}[/itex] into the vector y = [itex]\begin{bmatrix}y_1\\y_2\end{bmatrix}[/itex]

    The transformation can be re-written as:

    [tex]\begin{bmatrix}y_1\\y_2\end{bmatrix}=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}[/tex]

    Or, more succinctly, y=Fx


    Problem 1: Compute Fx For the following vectors x:

    a) x=[itex]\begin{bmatrix}1\\1\end{bmatrix}[/itex] b) x=[itex]\begin{bmatrix}1\\-1\end{bmatrix}[/itex] c) x=[itex]\begin{bmatrix}-1\\-1\end{bmatrix}[/itex] d) x=[itex]\begin{bmatrix}-1\\1\end{bmatrix}[/itex]


    My Results:

    a) Fx=[itex]\begin{bmatrix}3\\3\end{bmatrix}[/itex] b) Fx=[itex]\begin{bmatrix}-1\\-3\end{bmatrix}[/itex] c) Fx=[itex]\begin{bmatrix}-3\\-3\end{bmatrix}[/itex] a) Fx=[itex]\begin{bmatrix}-1\\3\end{bmatrix}[/itex]


    This is where I am unsure. The next step says "The heads of the four vectors x in problem 1 locate the four corners of a square in the [itex]x_1x_2[/itex] plane."

    I'm not sure I understand this: what does the " [itex]x_1x_2[/itex] plane" mean? I would have thought it means a plane in which [itex]x_1[/itex] and [itex]x_2[/itex] are the axes... But I can't see how this can work as [itex]x_1[/itex] just consists of the points 3, -1,-3 and 1 on the x axis, as far as I can see...

    I'd be very grateful if anyone could indicate where I'm going wrong..!
     
  2. jcsd
  3. Apr 16, 2014 #2
    [itex]x_1[/itex] and [itex]x_2[/itex] refer to your original vectors. The 4 points are [itex](\pm 1, \pm 1)[/itex]; these are the corners of a square.
     
  4. Apr 16, 2014 #3
    OK, thanks Chogg...
     
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