Please do Exercise 2.45 part A,B, and C and please show step by step and explain.if it is a subspace, do not need to prove. If not a subspace, give a counterexample 2.19 Example The picture below shows the subspaces of R³ that we now knowof the trivial subspace, lines through the origin, planes through the origin, andthe whole space. (Of course, the picture shows only a few of the infinitely manycases. Line segments connect subsets with their supersets.) In the next sectionwe will prove that R³ has no other kind of subspace, so in fact this lists them all.This describes each subspace as the span of a set with a minimal numberof members. With this, the subspaces fall naturally into levels — planes on onelevel, lines on another, etc.[x (1) + (9) + (x (1) + ² (1) ₁}0}{x|0}{y{x0+y{x1(+ (1) +- (69),+z 0}} {y1]} {y 1}{ [0]}+z 0} ✓2.45 Subspaces are subsets and so we naturally consider how 'is a subspace of'interacts with the usual set operations.(a) If A, B are subspaces of a vector space, must their intersection An B be asubspace? Always? Sometimes? Never?(b) Must the union AUB be a subspace?(c) If A is a subspace of some V, must its set complement V-A be a subspace?(Hint. Try some test subspaces from Example 2.19.)

2.45 a) If A, B are subspaces of a vector space, its intersection A∩B be a subspace. b) Must the…

✓ 2.45 Subspaces are subsets and so we naturally consider how 'is a subspace of' interacts with the usual set operations. (a) If A, B are subspaces of a vector space, must their intersection An B be a subspace? Always? Sometimes? Never? (b) Must the union AUB be a subspace? (c) If A is a subspace of some V, must its set complement V-A be a subspace? (Hint. Try some test subspaces from Example 2.19.)

✓ 2.45 Subspaces are subsets and so we naturally consider how 'is a subspace of' interacts with the usual set operations. (a) If A, B are subspaces of a vector space, must their intersection An B be a subspace? Always? Sometimes? Never? (b) Must the union AUB be a subspace? (c) If A is a subspace of some V, must its set complement V-A be a subspace? (Hint. Try some test subspaces from Example 2.19.)

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

Please do Exercise 2.45 part A,B, and C and please show step by step and explain.

if it is a subspace, do not need to prove. If not a subspace, give a counterexample

2.19 Example The picture below shows the subspaces of R³ that we now know
of the trivial subspace, lines through the origin, planes through the origin, and
the whole space. (Of course, the picture shows only a few of the infinitely many
cases. Line segments connect subsets with their supersets.) In the next section
we will prove that R³ has no other kind of subspace, so in fact this lists them all.
This describes each subspace as the span of a set with a minimal number
of members. With this, the subspaces fall naturally into levels — planes on one
level, lines on another, etc.
[x (1) + (9) + (x (1) + ² (1) ₁
}
0}
{x|0}
{y
{x0+y
{x1
(+ (1) +- (69),
+z 0}
} {y1]} {y 1}
{ [0]}
+z 0}

✓2.45 Subspaces are subsets and so we naturally consider how 'is a subspace of'
interacts with the usual set operations.
(a) If A, B are subspaces of a vector space, must their intersection An B be a
subspace? Always? Sometimes? Never?
(b) Must the union AUB be a subspace?
(c) If A is a subspace of some V, must its set complement V-A be a subspace?
(Hint. Try some test subspaces from Example 2.19.)

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