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    Linearity of Transformations; Basis and Dimensions for Four Subspaces

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    22. Which of these transformations is not linear? The input is v = (v1, v2).

    (a) T(v) = (v2, v1).
    (b) T(v) = (V1, v1).
    (c) T(v) = (0, v1).
    (d) T(v) = (0, 1).

    22. Without elimination, find dimensions and bases for the four subspaces for

    A = [0 3 3 3
    0 0 0 0
    0 1 0 1]

    and B = [1 1
    4 4
    5 5]

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    https://brainmass.com/math/linear-transformation/linearity-of-transformations-basis-and-dimensions-for-four-subspaces-120526

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    22.JPG
    (a) Yes.
    We consider u=(u1,u2), v=(v1,v2), and a scalar c, then we have
    T(u+v)=T(u1+v1,u2+v2)=(u2+v2,u1+v1)=(u2,u1)+(v2,v1)=T(u)+T(v)
    T(cu)=T(cu1,cu2)=(cu2,cu1)=c(u2,u1)=cT(u)
    So T is linear.
    (b) Yes.
    We ...

    Solution Summary

    Linearity of Transformations and Basis and Dimensions for Four Subspaces are investigated in this solution.

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