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Thread: Matrix associated to a linear map

  1. #1
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    Matrix associated to a linear map

    R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
    Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..
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    Quote Originally Posted by math.dj View Post
    R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
    Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..
    Choose non-zero vectors $\displaystyle v_1, v_2, v_3$ so that $\displaystyle v_1$ is in L (and since L has dimension 1, {$\displaystyle v_1$} is a basis for L) and {$\displaystyle v_2, v_3$} is a basis for W. You can do that because L directsum W= $\displaystyle R^3$. Applying T to $\displaystyle v_1$ you get $\displaystyle Tv_1= av_1+ 0v_2+ 0v_3$ because T maps L to itself. Applying T to either $\displaystyle v_2$ or $\displaystyle v_3$ gives $\displaystyle 0v_1+ bv_2+ cv_3$ because T maps W to itself.
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    W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..
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    Quote Originally Posted by math.dj View Post
    W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..
    If you really have no idea what the question is asking (and it appears from this that you don't) the best thing you can do is go to your teacher and ask for more explanation.
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  5. #5
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    Thx for the advice..
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