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Math Help - Lin. Transformation

  1. #1
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    Lin. Transformation

    NOTE: [a, b, c, d] <-- form is a vector and BOLD denotes a vector, unless referring to R.

    Suppose that T is a linear transformation from R^4 to R^3, defined by:

    T(e_1) = [1, 5, 3], T(e_2) = [2, 6, 2], T(e_3) = [3, 7, 1], T(e_4) = [4, 8, 0].

    1.) Find a standard Matrix A for the transformation.

    2.) Find T([5, -1, 3, 2])

    3.) Find an x such that T(x) = [-5, -9, 1]

    4.) Is T 1-1? If yes, explain why it is. If not, give 2 different vectors that map to the same vector.

    5.) Is T onto? If yes, explain why it is. If not, give a vector in the codomain that's not in the range.

    WORK (PLEASE CHECK):

    1.) The std. matrix A is:

    [[1, 2, 3, 4], [5, 6, 7, 8], [3, 2, 1, 0]] (3x4 matrix)

    2.) In order to find T([5, -1, 3, 2]), I would just take the matrix I formed in 1.) and multiply it by this vector, right?

    3.) To find an x such that T(x) = [-5, -9, 1], I'd augment this vector to my matrix in 1.), row reduce, etc.

    4.) NOT SURE!!

    5.) It's clearly not onto, right? Since, we're taking a transformation from R^4 to R^3...everything can't possibly go from something bigger to something smaller.
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  2. #2
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    Quote Originally Posted by Ideasman View Post

    4.) Is T 1-1? If yes, explain why it is. If not, give 2 different vectors that map to the same vector.
    .
    Is the kernel trivial? Meaning is the trivial element in R^3 the only element mapped into the zero vector?
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Is the kernel trivial? Meaning is the trivial element in R^3 the only element mapped into the zero vector?
    Um, I'm not sure. I don't even know what kernel means. I'm assuming the others are right..that method that is?
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  4. #4
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    Hmm, these look wrong, so let me show you what I have, and I'll try use LaTeX to make it look good:

    1.)

    \left[ \begin {array}{cccc} 1&2&3&4\\\noalign{\medskip}5&6&7&8<br />
\\\noalign{\medskip}3&2&1&0\end {array} \right]

    2.) For this one, I took my matrix above and multiplied it by:

    \left[ \begin {array}{c} 5\\\noalign{\medskip}-1\\\noalign{\medskip}3<br />
\\\noalign{\medskip}2\end {array} \right]

    to get:

    \left[ \begin {array}{c} 20\\\noalign{\medskip}56\\\noalign{\medskip}<br />
16\end {array} \right]

    3.)

    \left[ \begin {array}{ccccc} 1&2&3&4&-5\\\noalign{\medskip}5&6&7&8&-9<br />
\\\noalign{\medskip}3&2&1&0&1\end {array} \right]

    Then I row-reduced it:

     \left[ \begin {array}{ccccc} 1&0&-1&-2&3\\\noalign{\medskip}0&1&2&3&-<br />
4\\\noalign{\medskip}0&0&0&0&0\end {array} \right]

    Now I'm not sure what "an x" means...

    4.) Still need help on determining this.

    5.) I can see why it is NOT onto makes sense..not sure how to say why..


    Hope that helps make things look a little better.
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