is a Given vectors U₁, ..., up and w in V, show that w is a linear combination of u₁,..., up if and only if W linear combination of the coordinate vectorsu, 1 [u]. / * [W]B' To show the "if" part of the statement, assume that w ¹]... .. [ "P]B`` Write this linear combination below. [w] = Which of the following facts can be directly used to rewrite this linear combination? is a linear combination of the coordinate vectors OA. The coordinate mapping x-[x] is a linear transformation. B. The vectors u₁,..., up can each be written as a linear combination of the vectors in the basis B. O c. The coordinate mapping x-[x] is a one-to-one transformation. D. The basis B spans V.
is a Given vectors U₁, ..., up and w in V, show that w is a linear combination of u₁,..., up if and only if W linear combination of the coordinate vectorsu, 1 [u]. / * [W]B' To show the "if" part of the statement, assume that w ¹]... .. [ "P]B`` Write this linear combination below. [w] = Which of the following facts can be directly used to rewrite this linear combination? is a linear combination of the coordinate vectors OA. The coordinate mapping x-[x] is a linear transformation. B. The vectors u₁,..., up can each be written as a linear combination of the vectors in the basis B. O c. The coordinate mapping x-[x] is a one-to-one transformation. D. The basis B spans V.
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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![The linear combination should be rewritten as W
B
Since the coordinate mapping x→[x] is a(n)
B
maps to [w], namely the combination
above.
=
w which
which can be seen in the rewritten expression
transformation, there exists
This shows the "if" portion of the proof. The "only if" portion can be constructed by applying the same logic in
reverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb73e221-fc24-4f22-8ff5-c1a81fef495f%2Fa52f4ad5-8108-4129-bf52-5471cc95ffec%2F8lrqt0c_processed.png&w=3840&q=75)
Transcribed Image Text:The linear combination should be rewritten as W
B
Since the coordinate mapping x→[x] is a(n)
B
maps to [w], namely the combination
above.
=
w which
which can be seen in the rewritten expression
transformation, there exists
This shows the "if" portion of the proof. The "only if" portion can be constructed by applying the same logic in
reverse.
![Given vectors u₁.. up and w in V, show that w is a linear combination of u₁,
linear combination of the coordinate vectors [u, 1.....[u].
To show the "if" part of the statement, assume that
[₂]='
41]
[w] =
Which of the following facts can be directly used to rewrite this linear combination?
[₂] B
t[w]
up if and only if [ [W] B
is a linear combination of the coordinate vectors
Write this linear combination below.
O A. The coordinate mapping x→[x] is a linear transformation.
X
OB. The vectors u₁.. up can each be written as a linear combination of the vectors in the basis B.
O c. The coordinate mapping x→[x] is a one-to-one transformation.
OD. The basis B spans V.
is a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb73e221-fc24-4f22-8ff5-c1a81fef495f%2Fa52f4ad5-8108-4129-bf52-5471cc95ffec%2Fpzec41_processed.png&w=3840&q=75)
Transcribed Image Text:Given vectors u₁.. up and w in V, show that w is a linear combination of u₁,
linear combination of the coordinate vectors [u, 1.....[u].
To show the "if" part of the statement, assume that
[₂]='
41]
[w] =
Which of the following facts can be directly used to rewrite this linear combination?
[₂] B
t[w]
up if and only if [ [W] B
is a linear combination of the coordinate vectors
Write this linear combination below.
O A. The coordinate mapping x→[x] is a linear transformation.
X
OB. The vectors u₁.. up can each be written as a linear combination of the vectors in the basis B.
O c. The coordinate mapping x→[x] is a one-to-one transformation.
OD. The basis B spans V.
is a
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