2. Let CR4 where is subspace. Let be spanned by the vectorsV₁ = (1, 1, 1, 1), v2 = (1, -1, 2, 2), v3 = (1, 2, -3,-4)(a) Find an orthogonal basis for N.(b) Find an orthonormal basis for .3. Let V = P₂[t] be the space of polynomials in t of degree ≤ 2. Let 01, 02, 03 be the linearfunctionals on V defined by1$1(f(t)) = ["* f(t) dt, $2(f(t)) = f′(1), 43(f(t)) = ƒ(0)where f(t) = ao + a₁t+ a₂t² € P₂[t] and f' denotes the derivative of f. Find a basis {f1, f2, f3}of P2[t] that is dual to {01, 02, 03}.4. Let V be the vector space of square n x n matrices over R. Let Tr: V→ R; A Tr(A) =a11 + a22+...+ann be the trace mapping on V. Here A = (aij). Show that Tr is a linear map.

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Answered: 2. Let CR where is subspace. Let 2 be…

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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V is a 3-dimensional vector space with basis B = (vı, V2, v3). Let w = vi + v2 + v3, w2 = v2+V3, W3 = v1 +v2+2v3 and set B' = (w1, w2, W3), which is also a basis of V. a) Determine the change of basis matrix from B' to B, Pg-→B. b) Determine the change of basis matrix from B to B', P3¬B'.
In each part, find a basis for the given subspace of R', and state its dimension. a) All vectors of the form (a, b, 0, d). b) All vectors of the form (a, b, c, d) where a = b+d and c= 26 – d. c) All vectors of the form (a, b, c, d) where a = 26 = -c = 3d.
Solve a part only in 20 minutes plz i need it plz
Find an orthonormal basis of the plane x₁ + x₂ + x3 = 0
Given the basisof R consisting of the three vectors n = (-1,0,0) , v½ = (0,1,0) and V3 = (0,– 1,1) . Use Gram - Schmidt processto obtaina related orthonormal basisof R. %3D Then write (1,2,3) asa linear combination of the vectors in the new basis.
The coordinates for the vector v = (1,2,4) relative to the standard basis B = {(1,0,0), (0, 1,0), (0,0,1)} is [v] B = (1,2,4). The coordinates for v relative to the nonstandard basis B' = {(1, 0, -1), (2,1,-1), (-2, 1,4)} is [v]B¹ = (9,−1,3). Find the transition matrix P-1 from B to B' and show that this does change the coordinate of v relative to B to the coordinate of v relative to B'.
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Find an orthonormal basis for the span:
Consider the subspace S of the Euclidean inner product space R+ spanned by the vectors v₁ = (1,1,1,1), v₂=(1,1,2,4), v₂=(1,2,-4,-3). Find an orthogonal basis of S. O*((-.-3.1).(1.2.-4,-3).(4,2,1,1)) OB. © ³ {(1,1,2,4), ( − 1, − 1,0,2 A. -1,0,2), (12. 2.-3,1)} O C. None in the given list. OD. {(1,1,1,1),(1,1,2,4), (1,2,-4,-3) } OE {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) }
V = Span ({··]}) Find an orthonormal basis of R³ that contains the vector 圓

The third column of the change of basis matrix from the basis B= {u=(1,1,1) ; v= (1,0,1); w=(0,1,1 } to the basis B'= {r= (3,4,1); s= (3,0,1); t=(-4, -1, -2) is:

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2. Let CR where is subspace. Let 2 be spanned by the vectors V₁ = (1, 1, 1, 1), v2 = (1,-1,2, 2), v3 = (1, 2, -3,-4) (a) Find an orthogonal basis for N. (b) Find an orthonormal basis for .