Math 102 Homework 2
Due October 18 at 11:59pm PT
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(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.
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