3.Let T: R³ R³ be a linear operator and B = [V₁, V2, V3] a basis for R³. SupposeTOD-_TO-A_TO-7T(v₁)=T(v₂) =T(v3) =06]i.Determine whether w =5 is in the range of T0ii.Find a bases for R(T)iii.Find dim (N(T))

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Answered: Let T: R³ → R³ be a linear operator and…

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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For convenience, we write the coefficients uj,¿ in the vector {u} = (u₁, U₂,. where l - 1 + i + (N + 1)j. Q5 Ulg... ... 9 Then the second derivative matrix may be written as where is some operation on the one-dimensional second derivative matrix [D²] and the identity matrix [I] that produces the correct two-dimensional derivative matrix. [D²].x Urr (xi, Yi) = [I] [D²]{u}, ® x x Uyy (Xi, Yi) = [D²] [I]{u}, ‚ U(N+1)²)™ = (u0,0, U0,1, . . . , Wi‚j: For the moment, assume a 2 × 2 mesh. Show that the two dimensional derivative matrices are given by [D²] y = = [A] [B] D₁₁ D12 D² D2 11 21 D₁₁ D²₁ D%2 The result above can be extended to an N x N mesh. The operation the Kronecker product which is implemented in MATLAB using the kron() matrix [A] and a p × q matrix [B], it is defined as D²₁ D²21 11 D²₁₁ D₁2 D21 D22 D112 D 222 a₁1[B] ⠀ am1[B] ‚UN,N)T D²12 ain [B] : amn [B] on two matrices is called function. For an m × n
Find a vector parametric equation r(t) for the line through the points P = (4, −3, 4) and Q = (3, −6, 2) for each of the given conditions on the parameter t. (a) If 7(0) = (4, −3, 4) and 7(2) = (3, −6, 2), then r(t) = (b) If 7(6) = P and 7(8) = Q, then r(t) = = (c) If the points P and Q correspond to the parameter values t = 0 and t = -4, respectively, then F(t) =
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.

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Let T: R³ → R³ be a linear operator and B = [v₁, V2, V3] a basis for R³. Suppose 0 T(v₁) =_=A T(v₂): T (V3) = [33] i. Determine whether w = -- 0 ii. Find a bases for R(T) iii. Find dim (N(T)) is in the range of T