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One-to-one linear transformations

  1. Feb 24, 2013 #1
    Why is a linear transformation T(x)=Ax one-to-one if and only if the columns of A are linearly independent?

    I don't get it...
     
  2. jcsd
  3. Feb 24, 2013 #2

    micromass

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  4. Feb 24, 2013 #3
    Is there no alternative to insanely difficult wikipedia proofs?
     
  5. Feb 24, 2013 #4

    micromass

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    What does your textbook say? What is your textbook?

    Do they prove the rank-nullity theorem??
     
  6. Feb 24, 2013 #5

    jbunniii

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    T is one-to-one if and only if T(x) = T(y) implies x = y, if and only if T(x-y) = 0 implies x - y = 0, if and only if T(v) = 0 implies v = 0. But T(v) is a linear combination of the columns of A, so this says the only way to combine the columns of A to get zero is if the vector of coefficients (v) is zero. In other words, the columns of A are linearly independent.
     
  7. Feb 24, 2013 #6
    Ahh, thanks Jbunny. It makes perfect sense now!

    Micromass: Yes, it was, but the proofs in my book are written similarly to wikipedia - very tiresomely.
     
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