1. In the following Question (a-b), a function T: VW between vector spaces and two vectorsu and v in V are given.i.Find T(u) and T(v)ii.Is T(u + v) = T(u) +T(v)?iii.Is T a linear transformation?a) DefineT(ax³ + bx² + cx + d) ==[-a-b+c+db + ¹], Letu = −x³ + 2x² − x + 1 and v = x² − 1b) Define T: R³ →→ R² by T=(D=B²+2)₁ Let u 2 v=123TNT2

1. In the following Question (a-b), a function T: V---->W between vector spaces and two vectors u…

Answered: 1. In the following Question (a-b), a…

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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Q.3. Let R, (x) be a reflection of any vector x through the line along the vector u and P, (x) be a projection of any vector x onto the line along the vector u. Find the transformation RP for u =
Let e1, e2, èz be the standard unit vectors along the coordinate axes in R³. Let S and T be the linear transformations defind in R³. Show that if S(टे) = T(टे), S( हे ) %3D T(टे.), S(टेs) = T(ेs) then S(7) = T(7) for any 7 e R³.
1. Let T : R² → R³ be a linear transformation such that T(e1) = u1 and T(e2) = u2, where e1, e2 8 is the standard unit vectors of R² and u1 3 and u2 = 2. Then find T 1 -2 6.
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Suppose T: R2-→R³ is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(-3U-2V). 14 18 U = V = T(U) = -7 T(V) =-9 3 16 21 T(-3U–2V) = | 0
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Suppose T: R³-R² is a linear transformation. Let U, V and W be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(W). -3 3 2 3 ਪ = -3 -1 0 T(W) = 0 0 V = -3 W = | 3 - 1 T(I) = 0 T(V) = 0 -3
1. Let W be the line y = x in R². (a) Find a unit vector u on W. (b) Consider the linear transformation T: R² R² that projects each vector orthogonally onto W. Calculate the matrix for T which is uu¹. (c) Use that matrix to calculate T (d) Draw the vector [] [4] T[A] the line W, and the projection T
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