Let V and V' be vector spaces over a field F and B = {b,, b2,.., b,} be abasis for V. Then there exists a unique linear transformation, t: V-→ V' for any list b', b,',., b,'%3Dof vectors in V', such that1(b) = b,,i =1, 2,., n.

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Answered: Let V and V' be vector spaces over a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let N and N' be normed linear spaces with the some scalars and let T:N-N' be a linear T Transformation then prove that T is bounded if and T only if T is continuous.
4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
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…
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Let V and V' be vector spaces over a field F and B = {b,, b2,.., b,} be a basis for V. Then there exists a unique linear transformation, t: V-→ V' for any list b', b,',., b,' of vectors in V', such that 1(b) = b,, i =1, 2,.., n.