Let : R² R³ be a linear map such that /(1,2)=(1,1,2), and /(2,1)=(2,2,1). Thenwhich equalitie are correct?f(0,0) = (0,0,0).f(1, 1) = (1, 1, 1).f(-1,1)=(-1,-1,1).f(-1,-1)= (-1,-1,1).

Let V(F) and W(F) be two vector spaces over field F and f:V→W be a linear map. Then, a)…

Answered: Let : R³ → R¹ be a linear map such that…

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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Let G(u, v) = (2u + v, 5u + 11v) be a map from the uv-plane to the xy-plane. Find the image of the line v = 4u under G in slope-intercept form. (Use symbolic notation and fractions where needed.) y =
Determine whether each of the following maps from R? to R³ is linear. Prove your answers. (a) F((x,y)) = (x², y², x + y) (b) G( (x, y)) = (x+ y, 2x, 3y) (c) H((x, y)) = (x+1,y +2,0)
Let C be a piecewise curve consists of the following four line segments. First linesegment has vertices (0, 0) and (2, 0); the second segment has vertices (2, 0) and(2, 1); the third line segment has vertices (2, 1) and (0, 3); and the fourth linesegment has vertices (0, 3) and (0, 0). C is oriented counterclockwise. LetF(x, y) = ⟨x^2 − 2y, 2x⟩ be a vector field.(a) Sketch C accurately including the orientation.(b) Find the work done by F by moving a particle along the path C in the givendirection. Simplify your answer.
8. Decide which of the following sets are linearly independent in R. Justify your answer in each case. (a) X₁ = {(1,0), (0, 1)} CR² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} CR² (d) X₁ = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0, 1, 0), (1,1,1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} CR¹ (j) X10 = {(1,2,0,0), (0, 2, 3,0), (0, 0, -1, 1)} CR¹
Suppose T : C^2 → C^3 is a linear map so that T(1, 1) = (1, 0, 2) and T(2, 3) = (1,−1, 4). What is T(8, 11)?
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Show that (5,0,3) is in the span of S= {(1,1,0), (1,0,1), (1,1,1)}.

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Let : R³ → R¹ be a linear map such that /(1,2)=(1,1,2), and /(2, 1) = (2,2,1). Then which equalitie are correct? f(0,0) = (0,0,0). f(1, 1) = (1, 1, 1). f(-1,1)=(-1,-1,1). f(-1,-1)= (-1,-1,1).