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Math Help - matrix A for T with respect to the basis B.

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    matrix A for T with respect to the basis B.

    B is the basis {v1, v2, v3} of R3 where v1=[1, 1, 1], v2= [2, 3, 0], and v3= [-1, 2, -6] I need to find the matrix A for T with respect to the basis B. But somehow, I can't figure it out. It's given that T: R3--->R3 is a linear transformation and that T(v1)=v2, T(v2)=v3, and T(v3)=v1.

    Is T(v1)= [1, 0, 1]? I mean, how do you figure out what T(v) is given the values of v1, v2, and v3? It won't be [(2, 3, 0), (-1, 2, -6), (1, 1, 1)] because that would just be the change of basis right?
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    MHF Contributor FernandoRevilla's Avatar
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    If you know T(v_1)=v_2,T(v_2)=v_3,T(v_3)=v_1 then, the matrix of T with respect to the basis B=\{v_1,v_2,v_3\} is:

    A=\begin{bmatrix}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}  &{0}\end{bmatrix}

    you needn't more information about v_1,v_2,v_3 .


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    thanks, but I would really appreciate it if you tell me how you got the values.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Taurus3 View Post
    thanks, but I would really appreciate it if you tell me how you got the values.
    It is a well known theorem. You only need to transpose the coordinates of T(v_i) on B .


    Fernando Revilla
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    If this is the case, then what would be the value of ProjL([0, -1, 1]), ProjL([-1, 0, 1]), ProjL([2, 3]), and ProjL([-3, 2])?
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    Quote Originally Posted by Taurus3 View Post
    B is the basis {v1, v2, v3} of R3 where v1=[1, 1, 1], v2= [2, 3, 0], and v3= [-1, 2, -6] I need to find the matrix A for T with respect to the basis B. But somehow, I can't figure it out. It's given that T: R3--->R3 is a linear transformation and that T(v1)=v2, T(v2)=v3, and T(v3)=v1.

    Is T(v1)= [1, 0, 1]? I mean, how do you figure out what T(v) is given the values of v1, v2, and v3? It won't be [(2, 3, 0), (-1, 2, -6), (1, 1, 1)] because that would just be the change of basis right?
    You say you are told that "T(v1)= v2". and that v2= [2, 3, 0]. What would make you think that T(v1)= [1, 0, 1]?

    What you really need is that T(v1)= v2= 0v1+ 1v2+ 0v3, That T(v2)= v3= 0v1+ 0v2+ 1v3, and that T(v3)= v1= 1v1+ 0v2+ 0v3. You are using [a, b, c] to indicate vectors in the "standard" basis. If you use < a, b, c> to represent vectors in this basis, B, (so that v1= <1, 0, 0>, v2= <0, 1, 0>, and v3= <0, 0, 1>) then you are saying that T(v1)= T(<1, 0, 0>)= <0, 1, 0>, T(v2)= T(<0, 1, 0>)= <0, 0, 1>, and T(v3)= T(<0, 0, 1>)= <1, 0, 0>. What matrix does that?
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