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Math Help - Matrices...please help!!

  1. #1
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    Mar 2008
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    Matrices...please help!!

    Hello,

    I was wondering if someone could explain to me if I was given a linear transformation T defined by a 3 x 4 matrix whether it was "one to one"??

    I really confused when it says "T is one-to-one if and only if the equation
    T(x) = 0 has only the trivial solution."

    Thankyou so much greatly for your time.
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  2. #2
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    "T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution" is correct and means that the vector "x" is null. (For example, x=(0,0,0,0))

    So, to verify this statement, you have to solve a system of equations:

    T(x) = 0

    where the unknowns will be the components of "x".

    For example:

    T(x)=(x1+x2,x1)

    T is defined: T:R2->R2. The system to be solved is:

    (1) x1+x2=0
    (2) x1=0

    From (2), (1) is 0+x2=0 and x2=0. In this case, x=(0.0). This is the trivial solution.

    Well, now, suppose that N(x)=(x1-x2,x1-x2). Then, you have:

    (1) x1-x2=0
    (2) x1-x2=0

    So x1=x2. And x=(x1,x1). There you have multiple solutions to the system. Just asign a value for x1. Then, N is not one-to-one.

    In the case of a 3x4-matrix, you will have a system of 3 equations of 4 unknowns. You have to resolve:

    (1) equation1 = 0
    (2) equation2 = 0
    (3) equation3 = 0

    If you find more than one solution (you will), then it is not one-to-one.

    It is a simple help. If you want to go in-deep, there are multiple theorems to solve this problem in a different manner.

    Hope this helps.


    Federico.
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