TF. In parts (a)(i) determine whether the statement is true or false, and justify your

Chapter 8, Problem TF

(choose chapter or problem)

TF. In parts (a)(i) determine whether the statement is true or false, and justify your answer. (a) If T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1 and v2 in V and all scalars c1 and c2, then T is a linear transformation. (b) If v is a nonzero vector in V, then there is exactly one linear transformation T : V W such that T(v) = T(v). (c) There is exactly one linear transformation T : V W for which T(u + v) = T(u v) for all vectors u and v in V. (d) If v0 is a nonzero vector in V, then the formula T(v) = v0 + v defines a linear operator on V. (e) The kernel of a linear transformation is a vector space. (f ) The range of a linear transformation is a vector space. (g) If T : P6 M22 is a linear transformation, then the nullity of T is 3. (h) The function T : M22 R defined by T(A) = det A is a linear transformation. (i) The linear transformation T : M22 M22 defined by T(A) = 1 3 2 6 A has rank 1.