Question

help with p.1.13 please. thank you!

Group Name LAUSD Health N Vector Spaces P.1.9 Let V be an F-vector space, let wi, W2,...,W, EV, and suppose that at least one w; is nonzero. Explain why span{w1, W2,...,w,} = span{w; : i = 1,2,..., and W; 0). P.1.10 Review Example 1.4.8. Prove that U = {p EP3 : p(0) = 0) is a subspace of P3 and show that U = span{z.z.z). P.1.11 State the converse of Theorem 1.6.3. Is it true or false? Give a proof or a counterex- ample. P.1.12 In M,(C), let U denote the set of strictly lower triangular matrices and let W denote the set of strictly upper triangular matrices. (a) Show that U and W are subspaces of M.(C). (b) What is U+W? Is this sum a direct sum? Why? P.1.13 Let a,b,c ER, with a <c<b. Show that if € Cr[a,b] : f(c) = 0) is a subspace of CR[a, b]. P.1.14 Let vi. V2....,V, E C" and let A e M, be invertible. Prove that Av. Av2...., Av, are linearly independent if and only if v1, V2,...,V, are linearly independent. P.1.15 Show that (sin 2n: n = 1,2,...,) and {r : k = 0,1,2,...) are linearly independent sets in CRI-T,). 1.8 Notes A field is a mathematical construct whose axioms capture the essential features of ordinary arithmetic operations with real or complex numbers. Examples of other fields include the real rational numbers (ratios of integers), complex algebraic numbers (zeros of polynomials with integer coefficients), real rational functions, and the integers modulo a prime number. The only fields we deal with in this book are the real and complex numbers. For information about general fields, see [DF04). 19 Some Important

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2.1.13 Let abic GR. А2 В 2 , C Lauro] we have to show that, A-fff cp [ano] : fle) = 0} is a subspace of Let Or is the null mapping. so, OIRE A. ④ Let, e, and cq be two constants. La Take, fi and fz 6 A. then fr (c) = f(1) = 0 (a fit ez tz) () - a fi (c) + Czfeed - Oto=0 So, after to E A so A is a subspace of GR Cars]