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    Let T : R^7 to R^4 be a linear transformation and v1,v2,v3 be linearly dependent. Prove that T(v1), T(v2), T(v3) are linearly dependent in R^4.


    Give an Example of a non zero linear transformation S : R^3 to R^3 and an example of three linearly independent vectors v1,v2,v3 in R^3 such that the vectors S(v1), S(v2), S(v3) are linearly dependent.


    Totally lost on this. I get the basic idea of a linear transformation but have no idea how to prove the Above. Any easy methods?
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    (Original post by Zilch)
    Let T : R^7 to R^4 be a linear transformation and v1,v2,v3 be linearly dependent. Prove that T(v1), T(v2), T(v3) are linearly dependent in R^4.
    If v_1,v_2,v_3 are linearly dependent then there exist \lambda_1,\lambda_2,\lambda_3 such that

    \lambda_1v_1+\lambda_2v_2 + \lambda_3v_3 = 0

    Now apply T to this equation and use the defining properties of linear transformations.

    (Original post by Zilch)
    Give an Example of a non zero linear transformation S : R^3 to R^3 and an example of three linearly independent vectors v1,v2,v3 in R^3 such that the vectors S(v1), S(v2), S(v3) are linearly dependent.
    You want to find a transformation that takes something 3 dimensional and produces something that is either 1 or 2 dimensional. Just project onto a line or plane.

    Or, even more naively, a linear transformation is defined by its action on a basis so, pick the standard basis for v_1,v_2,v_3 and then simply pick any three vectors that are linearly dependent to map them too.

    Totally lost on this. I get the basic idea of a linear transformation but have no idea how to prove the Above. Any easy methods?
    Don't think about methods - just think about facts and definitions.
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    (Original post by Mark85)
    If v_1,v_2,v_3 are linearly dependent then there exist \lambda_1,\lambda_2,\lambda_3 such that

    \lambda_1v_1+\lambda_2v_2 + \lambda_3v_3 = 0

    Now apply T to this equation and use the defining properties of linear transformations.



    You want to find a transformation that takes something 3 dimensional and produces something that is either 1 or 2 dimensional. Just project onto a line or plane.

    Or, even more naively, a linear transformation is defined by its action on a basis so, pick the standard basis for v_1,v_2,v_3 and then simply pick any three vectors that are linearly dependent to map them too.



    Don't think about methods - just think about facts and definitions.
    Isn't one of the conditions for dependence to be preserved that a matrix be injective? If not is dependence never preserved?


    And for the second part, I tried but it was really just a guess. Let me know if its reasonably right.

    Let v1 = (1,2,3) v2= (2,1,3) v3 = (3,2,1)

    Let T be the linear transformation defined by T :R^3 to R^3 - T(a,b,c) = a

    Then, T(v1) = 1 T(v2) = 2 T(v3) = 3

    2T(v1) - 1T(v2) + 0T(v3) = 0

    Hence, there exists at least one non zero constant in the test for dependence.

    However,

    2T(v1) - 1T(v2) + 0T(v3) = 0
    = T(2v1-v2+0v3)

    2v1 - v2 + 0v3 = (0,3,3) 0

    The vectors are linearly independent before the transformation and dependent after.

    Does this answer the question reasonably or am i wrong somewhere in my example?
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    (Original post by Zilch)
    Isn't one of the conditions for dependence to be preserved that a matrix be injective? If not, is independence always lost?
    No, any linear transformation preserves linear dependencies. The hint I gave you for the first part of the exercise shows that.

    A linear transformation preserves all linear independencies if and only if it is injective but a non-injective linear transformation may still preserve some independencies.

    In any case, concentrate on trying to work with what you know and understand and can prove rather than just misremembering facts that you don't in any understand. In the second part of the question, you get carried away with this issue rather than answering the more basic one.... you need to learn and understand the definitions themselves first.


    (Original post by Zilch)
    And for the second part, I tried but it was really just a guess. Let me know if its reasonably right.
    Yeah, there is a problem with your answer - you need to go back and read the basic definitions of the terms you are using because you don't know the definition of linear independence which is probably why you don't understand your own answer and therefore are guessing.

    You didn't show that v_1,v_2,v_3 were linearly independent.

    Go and look up the definition of linear independence and then come back here and correct your own argument. In the meantime, to simplify that procedure, you will probably look at my previous post and see how you can simplify that by a much better choice of v_1, v_2 and v_3.
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    (Original post by Mark85)
    No, any linear transformation preserves linear dependencies. The hint I gave you for the first part of the exercise shows that.

    A linear transformation preserves all linear independencies if and only if it is injective but a non-injective linear transformation may still preserve some independencies.

    In any case, concentrate on trying to work with what you know and understand and can prove rather than just misremembering facts that you don't in any understand. In the second part of the question, you get carried away with this issue rather than answering the more basic one.... you need to learn and understand the definitions themselves first.




    Yeah, there is a problem with your answer - you need to go back and read the basic definitions of the terms you are using because you don't know the definition of linear independence which is probably why you don't understand your own answer and therefore are guessing.

    You didn't show that v_1,v_2,v_3 were linearly independent.

    Go and look up the definition of linear independence and then come back here and correct your own argument. In the meantime, to simplify that procedure, you will probably look at my previous post and see how you can simplify that by a much better choice of v_1, v_2 and v_3.
    Actually I didnt prove that v1,v2,v3 were linearly indep since I chose them knowing to be linearly independent. Other then that though, is the example sound?
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    (Original post by Zilch)
    Actually I didnt prove that v1,v2,v3 were linearly indep since I chose them knowing to be linearly independent. Other then that though, is the example sound?
    Then why did you write this:

    However,

    2T(v1) - 1T(v2) + 0T(v3) = 0
    = T(2v1-v2+0v3)

    2v1 - v2 + 0v3 = (0,3,3) 0

    The vectors are linearly independent before the transformation and dependent after.
    This makes no sense.

    Just read my last post anyway - if you go and look up what the basic terms mean then you will know for yourself whether your answer is right or wrong.

    If you can't be bothered to do that simple little thing, you will just remain at a sticking point and you will be spinning your wheels on simple questions like this.
 
 
 
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