Let T 1 and T 2 be linear transformations of U V and c be a scalar. Let transformations
Chapter 4, Problem 40(choose chapter or problem)
Let T 1 and T 2 be linear transformations of U V and c be a scalar. Let transformations T 1 + T 2 and cT 1 be defined by (T1 + T2)(u) = T1(u) + T2(u) and (cT1)(u) = cT1(u) (a) Prove that T 1 + T 2 and cT 1 are both linear transformations. (b) Prove that if T 1 and T 2 are matrix transformations defined by the matrices A1 and A2, then (T1 + T2) is defined by the matrix (A1 + A2), and cT1 is defined by the matrix cA1 (c) Prove that the set of transformations under these operations of addition and scalar multiplication is a vector space.
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