Let B = {b¡, following by filling in the blanks: b,} be a basis for a vector space V. You will be proving the ... If a subset {u,- 1,} is linearly dependent in V, then the set of coordinate vectors {{u, la[u,]a} is linearly dependent in R". ... You need only write the word for each blank on our quiz, but be organized so I can grade your work.
Let B = {b¡, following by filling in the blanks: b,} be a basis for a vector space V. You will be proving the ... If a subset {u,- 1,} is linearly dependent in V, then the set of coordinate vectors {{u, la[u,]a} is linearly dependent in R". ... You need only write the word for each blank on our quiz, but be organized so I can grade your work.
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
![Let B = {b¡,
following by filling in the blanks:
b,} be a basis for a vector space V. You will be proving the
...
If a subset {u,,
,} is linearly dependent in V, then the set of coordinate vectors
.... U
{{u, la [u,]} is linearly dependent in R".
You need only write the word for each blank on our quiz, but be organized so I can grade
your work.
(a) If set of vectors {u,, u,} is linearly dependent in V, then there exist scalars c,…,
(where at least one c; is non-zero),
(b) such that cu¡ +c,u, +
+c,u,=0, where the zero vector is in
(с) Вy the
of the coordinate mapping:
| qu, + ...
+c,u,=[qu,]a+
[c,u, =G[u,] ++c,
...+
u](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb055c63f-89a3-4c28-8011-cd1d10eeedf8%2Fdb04aba1-e2df-46f4-87fd-be8b2b74ce8a%2Fclkfllh_processed.png&w=3840&q=75)
Transcribed Image Text:Let B = {b¡,
following by filling in the blanks:
b,} be a basis for a vector space V. You will be proving the
...
If a subset {u,,
,} is linearly dependent in V, then the set of coordinate vectors
.... U
{{u, la [u,]} is linearly dependent in R".
You need only write the word for each blank on our quiz, but be organized so I can grade
your work.
(a) If set of vectors {u,, u,} is linearly dependent in V, then there exist scalars c,…,
(where at least one c; is non-zero),
(b) such that cu¡ +c,u, +
+c,u,=0, where the zero vector is in
(с) Вy the
of the coordinate mapping:
| qu, + ...
+c,u,=[qu,]a+
[c,u, =G[u,] ++c,
...+
u
![(d) (Note: u, la is a vector in
(e) Because the coordinate mapping is one-to-one, and since 0b, +Ob, + -+0__= 0,,
(1) [0, la =
the zero vector in
(g) Thus c[u,]+
·+c, u,= 0 , the zero vector in
(this is obtained by "taking
[ ]g of both sides of the equation in part (b)). These are the same c; scalars from part (a)!
(h) Therefore, the set of vectors
is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb055c63f-89a3-4c28-8011-cd1d10eeedf8%2Fdb04aba1-e2df-46f4-87fd-be8b2b74ce8a%2Fc8asif_processed.png&w=3840&q=75)
Transcribed Image Text:(d) (Note: u, la is a vector in
(e) Because the coordinate mapping is one-to-one, and since 0b, +Ob, + -+0__= 0,,
(1) [0, la =
the zero vector in
(g) Thus c[u,]+
·+c, u,= 0 , the zero vector in
(this is obtained by "taking
[ ]g of both sides of the equation in part (b)). These are the same c; scalars from part (a)!
(h) Therefore, the set of vectors
is
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