Q-3 (a) Suppose U = {(x, y, x+y, x – y, 2x) E F³ : x,y E F}Find a subspace W of F5, such that F5 = U OW.(b) In each part, determine whether the given vector%3D{x3 + x² + x + 1, x² + x + 1,2x - x² + x + 3 € P3 (R) is in the span of S =x +1 }.x³ + 2x² + 3x + 3 € Pa(R) is in the span of S = {x³ + x² + x + 1, x² + x + 1,x +1}.%3Di.ii.%3D (c)2/.va vector space over R with these operations? Examine all the properties and justify youranswer.(a1,a2) + (b, b2) = (a, + 2b,.a, + 3b,) and c(a,,a2) = (1,lu).et v E M2x2(F), be the vector space of all 2 x 2 matrices over the real number field.Define%3DW. = {(: ) eV:a, b,cEF}, W, = {(°, ) e V: a, b e F}Prove that W and W, are subspaces of V. and find the dimensions of W1,W2, W1+ W2andQ-2 (a) Let F be the field of complex numbers and let T be the function from F3 into F defined by7(X1, X2, X3) = (x1 - x2 + 2x3,2x1 + x2,-x1 – 2 x2 + 2x3)Verify that T is a linear transformation.%3Di.If (a, b, c) is a vector in F3, what are the conditions on a, b and c that the vector bein the range of T? What is the rank of T?What are the conditions on a, b and c that the vector (a, b, c) be in the null space ofT? What is the nullity of T?ii.iii.(b) Let T be the linear operator on R3 defined byT(x1,x2, X3) = (3x1, X1 - X2, 2x1 + x2 + x3 )Is T invertible? If so, find a rule for T-1 like the one which defines T.Suppose U = {(x, y, x+y, x-y, 2x) E FS: x,y E F}Find a subspace W of F, such that F5 = UOW.(b) In each part, determine whether the given vector%3Di.2x3-x² +x + 3 € P3(R) is in the span of S = {x3 +x2 +x + 1, x2 +x + 1,x+1}.

U=x,y,x+y,x-y,2x∈F5 : x,y∈ F U=span1,0,1,1,2,(0,1,1,-1,0) We have to find a subspace W of F5 such…

Answered: Q-3 (a) Suppose U = {(x, y, x+y, x- y,…

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Q-3 (a) Suppose U = {(x, y, x+y, x- y, 2x) E F5 : x,y E F} Find a subspace W of F5, such that F5 = UOW. %3D %3D

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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Q-3 (a) Suppose U = {(x, y, x+y, x – y, 2x) E F³ : x,y E F}
Find a subspace W of F5, such that F5 = U OW.
(b) In each part, determine whether the given vector
%3D
{x3 + x² + x + 1, x² + x + 1,
2x - x² + x + 3 € P3 (R) is in the span of S =
x +1 }.
x³ + 2x² + 3x + 3 € Pa(R) is in the span of S = {x³ + x² + x + 1, x² + x + 1,
x +1}.
%3D
i.
ii.
%3D

(c)
2/.
va vector space over R with these operations? Examine all the properties and justify your
answer.
(a1,a2) + (b, b2) = (a, + 2b,.a, + 3b,) and c(a,,a2) = (1,lu).
et v E M2x2(F), be the vector space of all 2 x 2 matrices over the real number field.
Define
%3D
W. = {(: ) eV:a, b,cEF}, W, = {(°, ) e V: a, b e F}
Prove that W and W, are subspaces of V. and find the dimensions of W1,W2, W1+ W2and
Q-2 (a) Let F be the field of complex numbers and let T be the function from F3 into F defined by
7(X1, X2, X3) = (x1 - x2 + 2x3,2x1 + x2,-x1 – 2 x2 + 2x3)
Verify that T is a linear transformation.
%3D
i.
If (a, b, c) is a vector in F3, what are the conditions on a, b and c that the vector be
in the range of T? What is the rank of T?
What are the conditions on a, b and c that the vector (a, b, c) be in the null space of
T? What is the nullity of T?
ii.
iii.
(b) Let T be the linear operator on R3 defined by
T(x1,x2, X3) = (3x1, X1 - X2, 2x1 + x2 + x3 )
Is T invertible? If so, find a rule for T-1 like the one which defines T.
Suppose U = {(x, y, x+y, x-y, 2x) E FS: x,y E F}
Find a subspace W of F, such that F5 = UOW.
(b) In each part, determine whether the given vector
%3D
i.
2x3-x² +x + 3 € P3(R) is in the span of S = {x3 +x2 +x + 1, x2 +x + 1,
x+1}.

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