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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Find a basis for the orthogonal complement of the subspace of R spanned by the vectors. V1 = (1,5, -5), v2 = (3, 14, 2), v3 = (1, 4, 12) The basis for the row space is W1 = ( 1 )
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
There are four vectors (1,a,b,c), (1,1,1,1), (1,2,3,1), (3,4,5,3). If the vector space (subspace) consisting of the linear combination of these vectors is T, find the condition that the number of vectors constituting the basis of T will be 3.
Let H span {(0,-1,-2), (-2, -4, 1), (3, 3, -1)}. A basis for the subspace HCR³ is { 2. }. vector
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.
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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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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 .