................................................./...............................................,,......../..............................................please show work as neat and complete as possible.thanks in advance................................................../...............................................,,......../.............................................. Include every possible detail in the solutions of all problems.Be sure to check your notation.Either prove that the given set S is a subspace of the vector space V or demonstrate that Sis not a subspace of V.In your proofs, do not combine steps.Any time you rely on a property of real numbers, indicate which property of real numbersI.you are using to go from one step to the next.Do not use linear combinations when proving S is a subspace.Instead, treat vector addition and scalar multiplication separately, the way we did it in class.To demonstrate that S is not subspace, show that one of the ten criteria for being asubspace fails to hold true.1)S = {a + bx + cx²: a + 2b – 3c = 1; a, b, c € R}2)S %3D {a + bx + сx?: а + 2b — Зс %3D 0%; а, b, с E R}3)= {[: "]:« € (-», 0]}I.Find a set of vectors S that spans the vector space V.Begin by parameterizing V; show your work.r2r – s1)V =3r + t: r, s,t E Zwith the usual operations from R°4r + 6s – t2)V =: 2a – d = 0, -2a + c – 2b + d = 0; a, b, c, d e Z }with the usual operations from M22

According to our guidelines we can solve only first three subparts of first question and rest can be…

Answered: 1) S = {a + bx + cx²: a + 2b – 3c = 1;…

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1) S = {a + bx + cx²: a + 2b – 3c = 1; a, b, cE R} 2) S = {a + bx + cx²: a + 2b – 3c = 0; a, b, c E R} 3)

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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................................................./...............................................,,......../..............................................

please show work as neat and complete as possible.

thanks in advance.

................................................./...............................................,,......../..............................................

Include every possible detail in the solutions of all problems.
Be sure to check your notation.
Either prove that the given set S is a subspace of the vector space V or demonstrate that S
is not a subspace of V.
In your proofs, do not combine steps.
Any time you rely on a property of real numbers, indicate which property of real numbers
I.
you are using to go from one step to the next.
Do not use linear combinations when proving S is a subspace.
Instead, treat vector addition and scalar multiplication separately, the way we did it in class.
To demonstrate that S is not subspace, show that one of the ten criteria for being a
subspace fails to hold true.
1)
S = {a + bx + cx²: a + 2b – 3c = 1; a, b, c € R}
2)
S %3D {a + bx + сx?: а + 2b — Зс %3D 0%; а, b, с E R}
3)
= {[: "]:« € (-», 0]}
I.
Find a set of vectors S that spans the vector space V.
Begin by parameterizing V; show your work.
r
2r – s
1)
V =
3r + t
: r, s,t E Z
with the usual operations from R°
4r + 6s – t
2)
V =
: 2a – d = 0, -2a + c – 2b + d = 0; a, b, c, d e Z }
with the usual operations from M22

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