{v1, v2, ..., vn} is an orthogonal set of vectors, then for any scalars{a1v1, a2V2, ..., anvn} is also an orthogonal set.Show that if Sd1, a2, ..., &n, the set T =

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Answered: Show that if S = {v1,v2,..., Un} is an…

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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Suppose V₁, V₂, V3 is an orthogonal set of vectors in R 5. Let w be a vector in Span(V₁, V2, V3) such that V₁ • V₁ = 42, V₂ · V₂ = 131, V3 · V3 = 9, W.V₁ = -126, w · V₂ = -655, w · V3 = 27, then w = V₁ + V₂+ V3.
Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so 7 and u are both members of the set R¹. We will say that and u are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, u) which asserts that n dimensional vector is almost the same as ū. Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to refer to the ith component of v. What set would i belong to in this case?
Consider the vectors u1 = (1, 2, 1), u2 = (0, 3, 2) and u3 = (−2, 1, 1). Determine whether the set S = {u1, u2, u3} is a spanning set for R3.
Let {v1, v2} be an orthogonal set of nonzero vectors, and let c1, c2 be any nonzero scalars. Show that {c\V1, C2V2}is also an orthogonal set. Since orthogonality of a set is defined in terms of pairs of vectors, this shows that if the vectors in an orthogonal set are normalized, the new set will still be orthogonal.
Suppose {v1, v2, ..., vp} is an orthogoal set of vectors. Show that {c1v1, c2v2, ..., cpvp} will also be an orthogonal set, for any choice of scalars c1, c2, ..., cp.
If {v1, v2, . . . , vn} is a linearly dependent set of vectors, prove thatone of these vectors is a linear combination of the other.
16, w be a vector in Span(v₁, V2, V3) such that = 11, V₂ V2 V1 V1 w v₁ = -55, w v₂ then w = • Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let . = . . = 46, V3 V3 184, w = v1+ . = V3 = 64, V2+ V3.
1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these vectors span V.
Let u = (1,4, 5), v = (-8, 2, 0), w = (3, 12, 17), which ones are orthogonal? u and v v and w u and w All of them
Suppose v1, v2, v3, 04, Ū5 and ve are vectors in R6 satisfying the following linear relation: 201 + 302 – Ủ4 + 205 - Which of the following must be true? (Select all that apply) span(v1, v2, v3, 4, 06) = span(71, v3, 74, 05, V6) span(v1, v2, v3, 74, 05) = span(v1, v2, v4, V5, 76) span(v1, v2, v3, v4, v6) = span(v1, 02, v3, v4, Ü5 , v6) span(v1, v2, V3, v6)= span(73, v4, V5 , V6) O span(v1, v2, v3, V4, 05) = span(72, 03, v4, 05, V6 )
Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in Span(V1, V2, V3) such that V1V1= 27, V2 V2 = 54, V3 V3 = 16, . . w.v₁ = -27, w v2 = 378, w V3: ₁+₂+ V3. then w = = 32, =

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Show that if S = {v1,v2,..., Un} is an orthogonal set of vectors, then for any scalars {qjv1,a2V2, ..., anVn} is also an orthogonal set. d1, a2, ..., &n, the set T =