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Math Help - proving linear transformation in R^3

  1. #1
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    proving linear transformation in R^3

    suppose T: R^3 -> R^3 is given by T(x1,x2,x3) = (x1,-x2,0).

    prove that T is a linear transformation. find the kernel, k, of T. Also, find dim K and dim (range T)

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  2. #2
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    Dear pandakrap,

    T is linear transformation if T(lambda*x) = lambda*T(x) and T(x+y)=T(x)+T(y).

    Now the first codition:
    Let x = (x1, x2, x3) so
    T(lambda*x) = (lambda*x1, -lambda*x2, 0) = lambda*(x1, -x2, 0) = lambda*T(x) --> it's true.

    second condition...

    y belongss to kernel if T(y) = 0.
    Now let y = (y1, y2, y3)
    T(y) = 0 --> (y1, -y2, 0) = (0, 0, 0)
    so the condition of kernel is y1=0 and y2=0 --> Kernel(T): (0, 0, t) where t is real. --> the kernel is one-dimensional.

    You know surely:
    Dim(R^3) = Dim(Kernel(T)) + Dim(Range(T)) .

    By the way, what is this T transformation doing? T affect as two linear transformation one after another: T = P_{xy}*R_y where P_{xy} is a projection in xy-plane and R_y is a reflection along y-axis. Why?
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