3. Consider the setl' = {x = (x1, x2, 13, .. .) E C* :i=1consisting of all absolutely summable infinite sequences of complex numbers.a) Show that e is a complex vector space.b) Show thati=1defines a norm on l.

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Answered: 3. Consider the set e = {x = (r1, x2,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Problem 1RQ

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(a) Let V be R², and the set of all ordered pairs (x, y) of real numbers. Define an addition by (a, b) + (c,d) = (a + c, 1) for all (a, b) and (c,d) in V. Define a scalar multiplication by k · (a, b) = (ka, b) for all k E R and (a, b) in V. . Verify the following axioms: (i) k(u + v) = ku + kv (ii) u + (-u) = 0
Which of the following is true about the subset T = {1+ 2x - x², 2 + 3x + x²,4 + 7x − x²} of the vector space R[x]<2 of polynomials of degree 2 or less? a) T is linearly independent. b) SpanT = R[x]≤2· c) T is linearly dependent. d) None of the choices in the list is correct.
Prove that ||ū – ū||< ||ū|| + ||ū|| where ū = (i, –2,1+ i) and w = (1,2 +i, –i).
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4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
Let ||. ||₁ and ||- |l2 be norms on vector space v if ||X→x||₁ → || x₂- Prove that ||. ||₁ and II. ||2 are equivalent x||2
Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n -vector y E R",3 a set of coefficients {t₁, t₂,... tn} with at least one t; #0 such that 1tixi = y. Prove that the coefficients t₁, t₂,.., t are unique.
3. Consider the set l' = {x = (x1, x2, x3,...) E C : i=1 consisting of all absolutely summable infinite sequences of complex numbers. a) Show that l1 is a complex vector space. b) Show that 8. ||x||1 = la| 2=1 defines a norm on l.
Let A be the set of all complex numbers z with principal argument θ ∈ [ π/4 , 5π/6 ] so that 1 ≤ |z| ≤ 2. Sketch A on the Argand plane.
Let V be an n-dimensional vector space over the field of complex numbers C. Showthat V over R has dimension 2n.
Q 06: For n≥0 , the set Pn of polynomials of degree at most n consists of all polynomials of the form p(t) = a0 + a1t + a2t2 +a3t3+......+ antn form a vector space. Where the coefficients a0, a1, a2, ..... an and the variable t are real numbers.
Consider R+ the set of all positive real numbers (here zero is not positive) equipped with the following operations: (i) u ⊕ v = uv, for any u, v ∈ R+ (ii) α ⊙ u = uα, for any α ∈ R and any u ∈ R+. show that R+ is a vector space (check the closure under the operations ⊕ and ⊙ and the 8 axioms).

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3. Consider the set e = {x = (r1, x2, x3, ...) E C* : i=1 consisting of all absolutely summable infinite sequences of complex numbers. a) Show that is a complex vector space. b) Show that ||x||1 =Elei| i=1 defines a norm on l.