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Homework Help: Ling. alg.

  1. Mar 26, 2006 #1
    Indicate whether each statement is always true, sometimes true, or always false.
    IF T: R^n --> R^m is a linear transformation and m > n, then T is 1-1
    Not sure to how prove this..
     
  2. jcsd
  3. Mar 26, 2006 #2

    HallsofIvy

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    T(x)= 0 for all x is a linear transformation. T(x)= x is a linear transformation.
     
  4. Mar 26, 2006 #3

    matt grime

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    The second hint doesn't help I'm afraid, Halls, T is a map from R^m to R^n and m>n.

    If T were injective, then its image is a subspace of what dimension?
     
  5. Mar 26, 2006 #4
    if it is injective and it goes from T^n to T^m, shouldn't the subpace be in the T^m ?
     
  6. Mar 27, 2006 #5

    HallsofIvy

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    What subspace are you talking about?

    If n< m, then you can think of Tn as a subspace of Tmp/sup]- think of adding 0's to the end of x.
     
  7. Mar 27, 2006 #6

    benorin

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    What does it mean for such a linear transformation to be one-to-one?

    Let [tex]T:R^n \rightarrow R^m[/tex], and suppose [tex]x_1,x_2\in R^n[/tex]. Then if [tex]x_1\neq x_2[/tex] implies that [tex]Tx_1\neq Tx_2[/tex], T is 1-1. Since T is linear, we have the last requirement becoming [tex]x_1\neq x_2\Rightarrow T(x_1-x_2)\neq 0[/tex]

    keep going...
     
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