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Math Help - Linear algebra matrices

  1. #1
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    Linear algebra matrices

    Let T: \Re^3 \rightarrow \Re^2 be a map defined as T(x_1, x_2, x_3) := (x_1 + x_2, x_1+ x_3)

    a) Find the matrix representation of T.
    b) Is T onto? Find the range of T.
    c) Is T one-to-one? Find the null set of T.

    Thank you!
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  2. #2
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    Quote Originally Posted by Linnus View Post
    Let T: \Re^3 \rightarrow \Re^2 be a map defined as T(x_1, x_2, x_3) := (x_1 + x_2, x_1+ x_3)

    a) Find the matrix representation of T.
    b) Is T onto? Find the range of T.
    c) Is T one-to-one? Find the null set of T.
    1) T(1,0,0) = (1,1) and T(0,1,0) = (1,0) and T(0,0,1) = (0,1).
    Thus the matrix A is \left[ \begin{array}{ccc}1&1&0\\1&0&1 \end{array}\right].

    2)If \bold{b}=(b_1,b_2) is in the range it means there is \bold{x}=(x_1,x_2,x_3) so that A\bold{x}=b.
    This means find all b_1,b_2 such that,
    \left[ \begin{array}{cccc}1 & 1 & 0 & b_1 \\ 1 & 0 & 1 & b_2 \end{array} \right]
    Can be reduced to a consistent system.

    3)The null set is A\bold{x} = \bold{0} for \bold{x} = (x_1,x_2,x_3).
    Thus, you need to solve,
    \left[ \begin{array}{cccc} 1& 1 & 0 & 0 \\ 1& 0 & 1 & 0 \end{array} \right]
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