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

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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 ...

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