From the the following transformations choose those that arelinear. A T : M 22 ®

Question

From the the following transformations choose those that arelinear. A
T : M22®R3 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = [a+2 bc+3 d, 2 b+2 c, -c+d] B
T : R2® R2 whereT([x1, x2]) = [-2 x1+x2, 0] C
T : R2® R3 whereT([x1, x2]) = [x1-x2, 3 x1x2,2 x1] D
T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û None of the above Plz show reasons behind ur answers. A
T : M22®R3 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = [a+2 bc+3 d, 2 b+2 c, -c+d] B
T : R2® R2 whereT([x1, x2]) = [-2 x1+x2, 0] C
T : R2® R3 whereT([x1, x2]) = [x1-x2, 3 x1x2,2 x1] D
T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û None of the above Plz show reasons behind ur answers. A
T : M22®R3 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = [a+2 bc+3 d, 2 b+2 c, -c+d] B
T : R2® R2 whereT([x1, x2]) = [-2 x1+x2, 0] C
T : R2® R3 whereT([x1, x2]) = [x1-x2, 3 x1x2,2 x1] D
T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û None of the above

Explanation / Answer

if the product of the components or entries is present,thenthe degree changes. that means linearity changes. in such a case , we cannot say it is a lineartransformation. if other than 0 constant is added to any of the components ,then also it is not a linear transformation. see that a constant is of degree 0 and so linearity ( degree1) fails. keeping these things in view, (1) [a+2 bc+3 d,2 b+2 c, -c+d] the term bc is the product of entries and so linearity failed.so, T is not a linear transformation. (2)  T([x1, x2]) =[-2 x1+x2,0] in the above given rule, i said other than 0. in this case 0 is given as a component. it can be viewed as 0* x1 or 0 * x2 which is of degree 1 andso, linearity is not disturbed. T is a linear transformation. (3) T : R2® R3 whereT([x1, x2]) = [x1-x2, 3 x1x2,2 x1] T is not a linear transformation. (4) T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û each term is linear and so T is a linear transformation. T : R2® R3 whereT([x1, x2]) = [x1-x2, 3 x1x2,2 x1] T is not a linear transformation. (4) T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û each term is linear and so T is a linear transformation. T : M22®M33 where T æ
ç
ç
è é
ê
ê
ë a b c d ù
ú
ú
û ö
÷
÷
ø = é
ê
ê
ê
ë 3 a 2 a+3 c a-2 d -c -d-a 0 3 c 2 b -2 a+d ù
ú
ú
ú
û

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