Show that every vector inT1 = span {(1,0,00); (0,1,0); (0,0,1)}are in also inT2 = span {(1,1,0); (0,0,1); (1,0,1)},i.e. show explicitly how i, in T2 correspondsto some given i, in T1.

Answered: Show that every vector in T1 = span…

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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1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list v1 - v2, v2 – v3, V3 - V4, V4 also spans V.
1.W.4 We'll work inside the vector space of polynomials in degree < 2, which is denoted P<2. Let P1 = 1, p2 = x + 2, and p3 = (x + 2)². Leť's think about Span{p1, P2, P3}. a) What polynomial is 3p1 + 2p2 – P3? b) I claim that a = a¡P1 + a2P2 + azp3 for some coefficients a1, a2, az in R. Find a1, a2, az. Hint: az = 0. c) I claim that a? = bịPi + b2p2 + b3p3 for some coefficients b1, b2, bz in R. Find b1, b2, bz. Hint: This problem isn't quite so cut and dry. Try to find three equations, one for each coefficient in the polynomial, and solve them for b1, b2, b3.
2. Find the projection projuv, where u = (1,3,-2) and v = (2,-1,3).
11. Assume u=(1,0,0,-1), vị=(3,2,7,3), v2=(2,1,3,2), and v3=(5,2,9,5). is in the orthogonal complement of W=span(v1,V2,V3). Show that u
6. Let 7₁, 7₂, .... that {V₁, V₂, . . . . . ., Ủn, } is linearly dependent. Un be vectors in a vector space V and let 7 be any vector in the span of S = 9 Ủi, U2,... 2 Un}. Prove
8. Show that {u,,,} is an orthogonal basis for R³. Then express x as a linear combination of the u's. 2₁ 2 -3 U₂ = 2 3 -=[3³] 5 ₁ and. =-3
2. In a vector space V with inner product (,), the vectors u and v are orthogonal and ||u|| ||v|| = 5. Determine the value of (u + 3v, 2u - 4v). = 3 and
Show that (5,0,3) is in the span of S= {(1,1,0), (1,0,1), (1,1,1)}.
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Show that every vector in T1 = span {(1,0,00); (0,1,0); (0,0,1)} are in also in T2 = span {(1,1,0); (0,0,1); (1,0,1)}, i.e. show explicitly how i, in T2 corresponds to some given i, in T1.