!Any finite dimension vector space with dim(X) is equal ton is isomorphism with Fتنشيط dows*TrueFalseIs that B' CBTrueFalseIf f (k_₁ α₁x₁) = -₁ αif(x₁) then=1i=1f(Σ=1 aix; ) = Σ+1 af(x;)TrueFalse

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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7. Prove that M2x2(C) is a complex vector space under the usual addition and scalar multiplication operations defined on 2 by 2 matrices.
Define Vector addition.
Let A = 0 -3 - 000 2 -1 - A basis for the null space of A is { }. vector
4. Let W be a subspace of a vector space V. We define a relation: V₁ ~ V₂ if V₁ V₂ € W. V1 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁ V2, V₂ V3 ⇒ V₁ ~ V3. 4b. Denote by V := = {x EV|x~v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W := {v + w|w€ W}. 4c. For v₂ EV, show: either V₁ V₂ = Ø or V₁ = V₂. Show: there are equivalence classes Va (a E A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vaj show: V₁ = V₂ ⇒ V₁ ~ V₂. (Remark. 4c hold for any equivalence relation on a set.) 4d. Show that the cardinality Va| = |VB| for any a, ß € A. Hint. Use 4b and show the bijectivity of the map: f: Va → VB (Va+W+Vg+w).
(a) On the set of people, (a, b) ∈R iff a and b both catch public transport (b) On the set of 2D vectors, (u, v) ∈R iff u * v = 0
Rather than use the standard definitions of addition and scalar multiplication in R, suppose these two operations are defined as follows. With these new definitions, is Ra vector space? ustify your answers. () ( Y. ) - 2) - * *. n* * 2) x. Y. 2) - (ox, cy, 0) O The set is a vector space. O The set is not a vector space because the associative property of addition is not satisfied. O The set is not a vector space because it is not closed under scalar multiplication. O The set is not a vector space because the associative property of multiplication is not satisfied. O The set is not a vector space because the multiplicative identity property is not satisfied. (6) . Ya. 72) - . V 2) = (0, 0. 0) c, Y. 2) - (ox, y, cz) O The set is a vector space. O The set is not a vector space because the commutative property of addition is not satisfied. O The set is not a vector space because the additive identity property is not satisfied. O The set is not a vector space because it is not closed under…
Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a unique non negative square root, that is, there is one and only one non negative operator N on V such that N^2=T.
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.
4. Define the following complex vector space = a € = (²x² = a +d=0} with the usual addition and scalar multiplication. (a) Find an appropriate value of r such that V~ Cr. (b) Give an explicit isomorphism T: V→ Cr. Make sure to prove that T is an isomorphism!
All the listed vector spaces are constructed over the field of real numbers R and equipped with the usual operations. In which case are the vector spaces isomorphic to each other? The correct answer is (b), or why?
Let V be an inner product space with dim V = 2, and suppose that L = {u+v, u+2v} is a linearly independent subset of V. Prove that if 1 is orthogonal to both u and v, then 1 = assume that V contains tuples. Also, be sure to use only results from our linear algebra course in your proof.) 0. (Note: You Imay not

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Any finite dimension vector space with dim(X) is equal to n is isomorphism with F dows but O True O False