Math 125Exercise - 3Linear Transformation:1. Show that the following maping f are linear:• f :R² → R² defined by f(r, y) = (x + y, x).f :R → R defined by f(x, y, z) = 2r – 3y + 4z.f: R? +R defined by f(r, y) = ry.• f:R? → R° defined by f(x, y) = (x + 1, 2y,r+ y).f: R' + R? defined by f(r, y, z) = (l#], 0).Linear Combination and Span of Vector Space and Sub Space:2. Show that the ry plane W = (r, y, 0) in Rº is generated by:(i)u = 2(ii)u =and v=and23. Show that that the vector u =-5 is a linear combination of the vectors v, =-33and ty =14. For which value of k the vector u =is a linear combination of the vectors3and vy =of the vectors v =and vz =-1Linearly In-dependency and Dependency5. Determine whether or not the following vectors i R are linearly dependent:(1, -2, 1), (2, 1, –1), (7, -4, 1)(1,2, –3), (1, -3, 2), (2, –1,5)(1, –3, 7), (2,0, –6), (3, –1, –1), (2, 4, –5)(2, –3, 7), (0,0,0)., (3, –1, –4)

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Linear Transformation: 1. Show that the following maping f are linear: • S : R² → R² defined by f(r, y) = (x + y, æ). • f:R → R defined by f(r, y, z) = 2r – 3y + 4z. • f: R →R defined by f(r, y) = ry. • f: R → R' defined by f(x, y) = (r + 1, 2y, r + y). S:R + R° defined by f(r, y, z) = (\r|, 0). %3D Linear Combination and Span of Vector Space and Sub Space: 2. Show that the ry plane W = (r, y, 0) in R is generated by: (i)u = |2 (ii)u = and v= 3 and 3. Show that that the vector u = -5 is a linear combination of the vectors v = 3 2 and ty = 4. For which value of k the vector u = is a linear combination of the vectors 3 3 and v = of the vectors v = and v = -2 Linearly In-dependency and Dependency 5. Determine whether or not the following vectors i R are linearly dependent: • (1, -2, 1), (2, 1, –1), (7, –4, 1) • (1, 2, –3), (1, –3, 2), (2, –1,5) (1, -3, 7), (2,0, –6), (3, –1, –1), (2, 4, –5) (2, -3, 7), (0,0, 0), (3, –1, -4)

Linear Transformation: 1. Show that the following maping f are linear: • S : R² → R² defined by f(r, y) = (x + y, æ). • f:R → R defined by f(r, y, z) = 2r – 3y + 4z. • f: R →R defined by f(r, y) = ry. • f: R → R' defined by f(x, y) = (r + 1, 2y, r + y). S:R + R° defined by f(r, y, z) = (\r|, 0). %3D Linear Combination and Span of Vector Space and Sub Space: 2. Show that the ry plane W = (r, y, 0) in R is generated by: (i)u = |2 (ii)u = and v= 3 and 3. Show that that the vector u = -5 is a linear combination of the vectors v = 3 2 and ty = 4. For which value of k the vector u = is a linear combination of the vectors 3 3 and v = of the vectors v = and v = -2 Linearly In-dependency and Dependency 5. Determine whether or not the following vectors i R are linearly dependent: • (1, -2, 1), (2, 1, –1), (7, –4, 1) • (1, 2, –3), (1, –3, 2), (2, –1,5) (1, -3, 7), (2,0, –6), (3, –1, –1), (2, 4, –5) (2, -3, 7), (0,0, 0), (3, –1, -4)

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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Question

Math 125
Exercise - 3
Linear Transformation:
1. Show that the following maping f are linear:
• f :R² → R² defined by f(r, y) = (x + y, x).
f :R → R defined by f(x, y, z) = 2r – 3y + 4z.
f: R? +R defined by f(r, y) = ry.
• f:R? → R° defined by f(x, y) = (x + 1, 2y,r+ y).
f: R' + R? defined by f(r, y, z) = (l#], 0).
Linear Combination and Span of Vector Space and Sub Space:
2. Show that the ry plane W = (r, y, 0) in Rº is generated by:
(i)u = 2
(ii)u =
and v=
and
2
3. Show that that the vector u =
-5 is a linear combination of the vectors v, =
-3
3
and ty =
1
4. For which value of k the vector u =
is a linear combination of the vectors
3
and vy =
of the vectors v =
and vz =
-1
Linearly In-dependency and Dependency
5. Determine whether or not the following vectors i R are linearly dependent:
(1, -2, 1), (2, 1, –1), (7, -4, 1)
(1,2, –3), (1, -3, 2), (2, –1,5)
(1, –3, 7), (2,0, –6), (3, –1, –1), (2, 4, –5)
(2, –3, 7), (0,0,0)., (3, –1, –4)

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