MLet f: R2 R³ be a linear map such that f(1,2)= (1, 1, 2), and f(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 and V' be vector spaces over a field F and let f: V→ V' be a linear map. Whichof the following statemens are correct?f is bijective if ker f = 0.f is surjective if Imf = V'.ker f is a subspace of V.f is injective if ker f = 0.

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Answered: O Let f: R2 R³ be a linear map such…

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 A(0,-2), B(0,4) (1) Find the set of all points P(x, y) such that m(PA)=2 m(PB) (2) Convert the equation in standard form and fully identify the equation.
[-1,2,-21. Let L be the line through the point Q-(-6, 9, -11) and direction vector Find the two distinct points R1 and R2 on L at distance 3 √10 to the point P-(-4, 2, -1). R1 =(0,0,0) R2=(0, 0, 0)
Let: R² R³ 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). ƒ(1, 1) = (1, 1, 1). Of(-1,1)=(-1,-1,1). Of(-1,-1)=(-1,-1,1).
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)
Consider these lines in three-space. l₁ : [x, y, z] = [5, -3, 6] + [-4, 1, -2] l₂: [x, y, z]= [3, -8, -6] + s[-1, 3, 5] 3: [x, y, z]= [4, 0, −1] + v[−2, 1, 2] a) Does (-3, -1, 2) lie on line ₁? b) Determine if line ₁ intersects line 2. c) Determine if line , intersects line 3.
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
(c) Eliminate B and a from the relation, y = Bcos(wx + a). In which w is a parameter (not to be eliminated).
Decide whether Y1 (t) = t +3, Y2(t) = t? – t, y3 =t+3 are linearly independent or linearly dependent. In the latter case find a linear relation among them.

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O Let f: R2 R³ be a linear map such that f(1,2)= (1,1,2), and f(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). O ƒ(-1, -1) = (-1, —1, 1).