Math 102 Homework 2 Due October 18 at 11:59pm PT Please neatly write your solutions in complete sentences and label them clearly. To submit your work, log in to Gradescope with your @ucsd.edu email (either directly or through Canvas). Please make sure the scan is clear and prepare early since uploading takes some time. (1). Let V be a vector space. Suppose v1 , v2 , v3 , v4 is a basis of V . Prove that v1 ` v2 , v2 ` v3 , v3 ` v4 , v4 is also a basis of V . (2). Find a basis of the following subspace of R3 : tpx, y, zq P R3 : x “ 5zu (3). Recall that P4 denotes the subspace of real coefficients polynomials of degree ď 4. (a). Let U “ tf P P4 : f p0q “ 0u. Find a basis of U . (b). Let W “ tf P P4 : f p6q “ 0u. Find a basis of W . (c). Let Z “ tf P P4 : f 2 p0q “ 0u. Find a basis of Z. (4). Suppose U and W are both five-dimensional subspaces of R9 . What are the possible dimensions of U X W ? (5). Suppose T : R3 Ñ R is defined by px, y, zq ÞÑ x ` 2y ` 3z. (a). Verify that T is a linear map. (b). Find a basis of kerpT q. (6). Give an example of a linear map T : V Ñ W such that dim kerpT q “ 3 and dim impT q “ 2. (7). Let T : P4 Ñ P4 be the differentiation operator: f ÞÑ f 1 . Let U “ tf P P4 : f p0q “ 0u. (a). Is T injective? Is T surjective? Explain why. (b). Calculate dim kerpT q and dim impT q. (c). Show that T : U Ñ P4 is injective. 1