Please solve B27. The answer should be hand written.

For Problems B19-B28, find a basis for the range and a Bi T : Mozz(R) - Mzx2(R) defined by T(A) = AT basis for the nullspace of the linear mapping and verify the Rank-Nullity Theorem. For Problems B8-B12, determine whether the mapping is B19 L : R' - P2(R) defined by linear L(a, b, c) = a + bx + cx B8 L : P,(R) - R defined by L(a + bx + cx]) = a - 2b B20 L : P, (R) - R' defined by L(a + bx) = |a+ b a - b B9 M : P,(R) - R defined by M(a + bx) = b -a B21 L : P2(R) - P, (R) defined by B10 N : R' - M2x2(R) defined by L(a + bx + cx) =b+2ax x3 B22 L : M2x2(R) - R3 defined by & (|= =]) - By L: Mza(R) - Mza(R) defined by L(A) = Al B25 L : R' - Mzx2 (R) defined by L(X1. *2, X1) = *1 + 212 -x14 2x3] x2 + x3 X1+2x2 B12 T : Mba(R) - P,(R) defined by Ta - 6) = a+ (a+ d)x+ 16+ clix B24 Let D be the subspace of M2x2(R) of diagonal matrices; L : D - P2(R) defined by For Problems B13-B18, determine whether the given vec- 2 (8 8) = a+ (a + bix+ bx] I tor y is in the range of the given linear mapping L : V - W. If it is, find a vector x e V such that L(x) = y. B25 L : Max2(R) - M2x2 (R) defined by BY3 L : R3 - R3 defined by 1 2) = c-d d-a -X1 - 2x2 = 2 x1 + *3 B26 L : M2x2(R) - P2(R) defined by -2x1 + x2 - 2x3 ( e 2) = ( a + b ) + (c + 0)x] Section 4.5 Exercises 283 BX T : Muz(R) - M2x2(R) defined by T(A) = AT B31 L : R'(R) - P, (R) defined by L(a, b, c) = (a + b+ 2c) + (a +2cir+ (a+26 + ckx B28 L : P2(R) - M2xz(R) defined by M : P(R) - R' defined by L(a + bx + cx)= -0- 2c 2b-c M(a + bx + ex ) = (-4a +36 + 2c,a -b.2a -b-c) -20 + 20 -2b -c For Problems B32-B34, construct a linear mapping For Problems B29-B31, determine whether L and M are L ; V - W that satisfies the given properties