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# (Solved): Show that: (a) the followR^(3);(b)Tv=(1;2;1;)w=(3;1;1)T_(1)(x_(1);x_(2);x_(3))=(x_(1),x_(2);x_(1)+2x ...

Show that: (a) the follow

R^(3);(b)Tv=(1;2;1;)w=(3;1;1)T_(1)(x_(1);x_(2);x_(3))=(x_(1),x_(2);x_(1)+2x_(3);x_(2),x_(3))T_(2)(x_(1);x_(2);x_(3))=(x_(1)+2;x_(2)1;x_(3)+1),T_(3)(x_(1);x_(2);x_(3))=(x_(1)x_(2);x_(1)+x_(3);x_(2)x_(3))T:R^(3)!R^(3)T(1;,1;1)=(3;,1;1),T(2;1;2)=(2;,2;1)T(1;0;1)=(1;1;0)T(3;1;2)T=D_(x)^(2)2D_(x)C^((2))[a;b]C[a;b]T(u)=u^((2))(x)+u^((1))(x)u2C^((2))[a;b]TC^((2))[a;b]C[a;b]TP_(2)P_(2)P_(n)nT(x+2)=x^(2)\times 2Tx^(2),x=2x^(2),x+2,Tx^(2)+1=x^(2),3x. Find Tx^(2)+2x,4; T:P_(2)!P_(1),Ta_(0)+a_(1)x+a_(2)x^(2)=a_(1)x^(2),2a_(2)xT(x_(1);x_(2);x_(3))=(x_(1)x_(2);x_(1)+x_(3);x_(2)2x_(3)) is a linear in the space R^(3);(b) use the function T to nd (b) the image of v=(1;2;1;) and (c) the preimage of the w=(3;1;1); Determine whether the maps T_(1)(x_(1);x_(2);x_(3))=(x_(1),x_(2);x_(1)+2x_(3);x_(2),x_(3)); T_(2)(x_(1);x_(2);x_(3))=(x_(1)+2;x_(2)1;x_(3)+1),T_(3)(x_(1);x_(2);x_(3))=(x_(1)x_(2);x_(1)+x_(3);x_(2)x_(3)) are a linear or nonlinear transformations; Let be a linear transformation T:R^(3)!R^(3) such that T(1;,1;1)= (3;,1;1),T(2;1;2)=(2;,2;1), and T(1;0;1)=(1;1;0). Find T(3;1;2); Let be T=D_(x)^(2)2D_(x) the transformation from C^((2))[a;b] into C[a;b] de ned by T(u)=u^((2))(x)+u^((1))(x) for u2C^((2))[a;b] : Show that T is linear transformation from C^((2))[a;b] into C[a;b]; Let T be a linear transformation from P_(2) into P_(2). Here, P_(n) denotes the space of polinomials with degree n such that T(x+2)=x^(2)\times 2, Tx^(2),x=2x^(2),x+2,Tx^(2)+1=x^(2),3x. Find Tx^(2)+2x,4; Find the kernel and range of the following transformation T:P_(2)!P_(1),Ta_(0)+a_(1)x+a_(2)x^(2)=a_(1)x^(2),2a_(2)x

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