MATH 219 Final, Fall 2003

Math 219 ban ey Final Exam OF cUnoT January 23, 2004 Time: 90 minutes All vector spaces are assumed to be finitely generated. Let T: U > V and S : VY ~ W be linear transformations of vector spaces. Prove that the composed map ST : U > Wis a linear transformation. Let T: ¥ > Woe a linear transformation of vector spaces. Prove that the map T is injective if and only if ker T = 0. Let T : V > W be an injective linear transformation of vector spaces and let fai, . . . Gn}, m 2 1, be a linearly independent subset of V. Prove that {T(a.), .. ., T(@n)} is a linearly independent subset of Y/ Let T: ¥ > V be an injective linear transformation from the vector space V to itself. Prove that T is an isomorphism. ( Hint: use the result in problem 3 or recall the formula dim VY = dim ker T + dim im T and exercise 2). Let VY, and VY; be subspaces of a vector space V such that V¥; + 4 = V and V, oO bs = 0. Prove that dim V = dim VY, + dim V3. Let V be the vector space of all polynomials over R of degree <n, n 2 1 and let T : ¥ — V be the operation of forming derivatives. Write down the n x n matrix A of T relative to the ordered basis {1,x,...,.x"~'}.