5. Let T: IR'>IR bé the livenr trainstormationde fined by T(X,X]=(xi+3Xz,?x;-S X2).B= {u, Uz} and B' ={,,Vz} whereU,= C6], uz=[?], v,%3D= V2-).%3Dal find the matrix for T velafive tobasis B.b) use the result in part c(a) to Computethe mectrix T helutive to basis B'.

A linear transformation is a function from one vector space to another that respects the underlying…

Answered: 5. Let T: IR'Ie bé the livear…

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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9. Consider the orthonormal basis S for R' S = {v₁,v₂,v,} where v₁ = (0,1,0) S= v, -(-3.0,) and v, - (3.0.3). Let u = (1,0,4). Find (2), = 8 19 A (u), -(0,-3,1) = 8 19 B (u), - (1,3,7) = 8 19 C (u), = (0,3, 1/2) D No correct answer E (u) = (0,8,19)
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4. Let T-RR' be the lineor trons.form such that T (xiyiz)=(yt7, X+y, X+Z), By considering matrix representating T with re spect to basis you like, answer the following questionss any * T is inver tible linear transformatione Yor N? 3. For every b ER, the equation T(P)=b is solvable Y or N ? vector solution. * The equation TR)= has nonzero Y or N? for As ģ : 7. let S:RR be the by s? ) = A? Now 4 lincar trous formation defined the the composition So T-R R and auswer following pue. Consider S.T is c înver tible line ar trans formation, Y or NS an A for every BER the eguation S. T(?)=B is solvable the only Solution of the equation. S. TQ)= 8 vector is N? or
Consider a linear model with decislon boundary w" + b stands for vector transpose, - 2, w and a are both column vectors, What will be the classification, +1 or 0 for a 3D feature space, where (1,2,3)" (1.c., w is a column vector) and b 1, of a test example defined by a = (1, –1, –1)" ? O+1 .1
1. W is the plane passing through origin and orthogonal to i = Find the projection matrix for W.
Let CER" be a matrix such that xCx > 0 for all nonzero vectors & Prove that for k= 1,...,n, the leading principal submatrix C, satisfies y all nonzero vectors y ERK. Consequently, each C is nonsingular. R" Cky > 0 for
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
Answer the following questions: (a) () Find the 12-norms of the vectors, x = (1,2,3)™ and y = (1,2,3)T and y = (9, 11, 13). (b) ma Show that the set, ( is orthonormal. T T 5-{(2,-1,0)", (1,2-5), (2.4.2)"} 2, 4, = S = T 1 30 √24 (c) i- Let f(x) sin x and consider co = 4, x1 9 and x2 = = after rounding. Now, Find the polynomial using Least Square Approximation for n = 1. -6. For f(x) take up to 3 decimal places
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
6. Determine the matrix of the linear mapping defined by first mirroring the space vectors in the line with equation (x, y, z) = (2-2+, -2+2+, 1-t) and then rotating 풍 them by the angle II around the x-axis in the positive direction.
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5. Let T: IR'Ie bé the livear trainstormation de fined by T(X,X]<(xi+3Xz,?x;-S X2). and B= {u, Uz} and B'={v,Vz} where %3D %3D alfind the matrix for T vclafive to basis B. b) use the result. in part (a) to Compute the meitrix T relative to basis B'.