Let T:R2 Rightarrow R3 be defined T([x1 x2]) = [x1+2x2 -x1 0] Find the matrix [T

Question

Let T:R2 Rightarrow R3 be defined T([x1 x2]) = [x1+2x2 -x1 0] Find the matrix [T]B',B with respect to the bases B = {u1, u2} and B' = {v1, v2, v3}, where u1 = [1 3], u2 = [-2 4], v1 = [1 1 1], v3 = [3 0 0] Verify that formula 4a holds for every vector [T]B',B = [[T(u1)]B'|[T(u2)]B'|...[T(un)B'] x = [x1 x2] in R2.

Explanation / Answer

x.P5th="52" style="mso-width-source:userset;mso-width-alt:1901;width:39pt">LET U=[X1,X2] TO STD.BASIS AND U=[Y1,Y2] TO THE GIVEN BASIS . V=[Z1,Z2,Z3] TO STD. BASIS AND V=[T1,T2,T3] TO THE GIVEN BASIS GIVEN THAT L[U]=L[X1,X2]=L[X]=A*[X1,X2]=[X1+2X2,-X1,0]=[Z1,Z2,Z3]=Z L[X]=A*X=Z…………………………………………………………1 WHERE A IS TRANSORM MATRIX WRT STD.BASES…SO WE HAVE C D * X1 = CX1+DX2 =Z1= X1+2X2 M N X2 MX1+NX2 =Z2= -X1 P Q PX1+QX2 =Z3= 0 HENCE WE GET C=1…..D=2…..M=-1….N=P=Q=0 A= 1 2 -1 0 0 0 NOW TO FIND TRANSFORM MATRIX B SO THAT   L[U]=L[Y1,Y2]=L[Y]=B*[Y1,Y2]=[T1,T2,T3]=T L[Y]=B*Y=T………………………………………………………….2 U=[X1,X2] TO STD.BASIS AND U=[Y1,Y2] TO THE GIVEN BASIS.SO . TRANSFORM MATRIX FOR U WRT NEW BASIS IS G WHERE G= 1 -2 3 4 HENCE WE HAVE

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