(a.2) Let V be the vector space of polynomials over K of degreen-1 andf: V -> Vthe map defined by derivative:f(u(r))u(x) u(x)e V.Show that f"0 and describe all invariant subspaces of V. Is V decomposable?(Hint: Consider two cases: K contains Q (so all nonzcro integers are nonzeroin K), or K contains Z, where p > 1 is a prime (so p = 0 in K and i0 forevery integer i with 0 < i < p)).

Given that V be the vector space of polynomials over K of degree less than equal to (n-1) and

Answered: (a.2) Let V be the vector space of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Let V be the vector space of all polynomials over C with degree < n. By giving a basis for V, show that V is a finite dimensional vector space over C. What is the dimension of V? e Let V be the set of all continuous functions form (0, 1] -→ R. We know that it is a vector space over R. Is V a finite dimensional vector space over R? Justify your answer,
7. (**) Let V be a vector space over C of dimension n. Let X be the vector space of linear transformations from V to V; denote by 0 the zero map on V, which is the "zero vector" in the vector space X. Also, X contains the identity map I: V→ V. Let L: V→V be a linear transformation (so that I can be viewed as a vector in X). Further, L²= LoL EX and L³ = Lo Lo € X, etc. We define Lº := I. If q(t) is a polynomial with coefficients in C, then q(L): V → V is defined to be the following linear transformation: q(L) = apI+a₁L + a₂L²+...+ am L™ where q(t) = ao+a₁t+...+amt™ that is q(L) is just a linear combination of the powers of L, so q(L) is a "vector" in X. Assume that L 0. Prove the following: a) Using the fact that X is a vector space of dimension n² over C, show that the set of powers of L must be linearly dependent. b) Using (a), show there exists a polynomial q(t) such that q(L) = 0, the zero map. c) Show that there exists a least integer k such that there is a monic polynomial q(t) of…
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
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2) injective Let V be a real vector space. Prove that R linear map described below is V → Vc v→ 10v (This lets us "think" of V as being a subspace of Vc)
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