✓ 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}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F20a5ad0b-e444-45da-8840-dc7449d7f6f5%2Fp0sfuyk_processed.png&w=3840&q=75)
Transcribed Image Text: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}
Transcribed Image Text:✓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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