8. Consider the following linear transformation T P- -> P2 defined by T(p(x)) = p(0)x +...

  

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8. Consider the following linear transformation T P- -> P2 defined by T(p(x)) = p(0)x + p(1) where p(x) = Po + Pix+P2x. (a) Find a basis for the kernel and the range of T. (b) Find the rank and nullity of T. (c) Find [T]B with respect to the standard basis B = {1, x, x}. 9. Suppose T is defined by T(x) = Ax where -3 2 A = 0 -4 3 -9 k (a) Find all values of k that will make T a one to one linear transformation. (b) Find all values of k for which R(T) is R. (c) Show that T is an isomorphism and find T-. 10. Let T: P2P be a linear transformation defined by T (ax + bx + c) = ax + (a - 2b) x + b (a) Determine whether p(x) = 2x-4x+6 R(T). (b) Find a basis for R(T). 11. Define a linear transformation T : M2x2 M2x2 by T(A) = [11] 4 (a) Determine if T a 1-1 linear transformation. (b) Find the rank and nullity of T. - 0 A-A A[11]. 12. Determine whether the linear transformation T: R3 R defined by T is an onto linear transformation. I = 2x + 3y - z -x+y+3z x+4y+ 2x 13. Let T R. : R be a linear transformation defined by -2 0 1 T (v) = 1 -1 -1 .v. 0 1 0 (a) Show that T is an isomorphism (b) Find T- if it is exist. 8. Consider the following linear transformation T P- -> P2 defined by T(p(x)) = p(0)x + p(1) where p(x) = Po + Pix+P2x. (a) Find a basis for the kernel and the range of T. (b) Find the rank and nullity of T. (c) Find [T]B with respect to the standard basis B = {1, x, x}. 9. Suppose T is defined by T(x) = Ax where -3 2 A = 0 -4 3 -9 k (a) Find all values of k that will make T a one to one linear transformation. (b) Find all values of k for which R(T) is R. (c) Show that T is an isomorphism and find T-. 10. Let T: P2P be a linear transformation defined by T (ax + bx + c) = ax + (a - 2b) x + b (a) Determine whether p(x) = 2x-4x+6 R(T). (b) Find a basis for R(T). 11. Define a linear transformation T : M2x2 M2x2 by T(A) = [11] 4 (a) Determine if T a 1-1 linear transformation. (b) Find the rank and nullity of T. - 0 A-A A[11]. 12. Determine whether the linear transformation T: R3 R defined by T is an onto linear transformation. I = 2x + 3y - z -x+y+3z x+4y+ 2x 13. Let T R. : R be a linear transformation defined by -2 0 1 T (v) = 1 -1 -1 .v. 0 1 0 (a) Show that T is an isomorphism (b) Find T- if it is exist.