Let B={b,,..., b,} be a basis for a vector space V. You will be proving the following by filling in the blanks: If the set of coordinate vectors u,la: u, a is linearly dependent in R", then the subset u,} is linearly dependent in V. You need only write the word for each blank on our quiz, but be organized so I can grade your work. (a) If the set of vectors {u,a: [u,]a} is linearly dependent in (b) then there exist scalars c,- (where at least one c; is non-zero), (c) such that = 0 the zero vector in R" (d) By the of the coordinate mapping: „[u,] =[c«]la+ +[c,u,]a=[qu, + +c, +c, ...+ pp (e) (Note: c,u, is a vector in (f) Because 0, R" = [0, ]a' we can see that from above and part (c) that we have [gu, + ..+c,u,] =[0, ]g• Initial if you agree

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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(g) Because the coordinate mapping is
[qu, + ..+c,u, ] =[0,]a implies qu, + .+c,u, = 0, .
...+ C.
v ]B
(h) Therefore, the set of vectors
is
Transcribed Image Text:(g) Because the coordinate mapping is [qu, + ..+c,u, ] =[0,]a implies qu, + .+c,u, = 0, . ...+ C. v ]B (h) Therefore, the set of vectors is
Let B={b,,...,b,} be a basis for a vector space V. You will be proving the
following by filling in the blanks:
If the set of coordinate vectors {[u,la u, a is linearly dependent in R", then the subset
{u,u,} is linearly dependent in V.
You need only write the word for each blank on our quiz, but be organized so I can grade
your work.
(a) If the set of vectors {u,a
·[u,] is linearly dependent in
(b) then there exist scalars c,,
(where at least one c, is non-zero),
(c) such that
= 0
the zero vector in R" .
(d) By the
of the coordinate mapping:
G[u,]a+
+Cpup
си.
...+C
u
+
+
+ -..
(e) (Note: c,u, is a vector in
(f) Весause 0.
R"
=|0, la, we can see that from above and part (c) that we have
[qu, +.+c,u,]=[0, la: Initial if you agree
+c̟u,
pp ]B
Transcribed Image Text:Let B={b,,...,b,} be a basis for a vector space V. You will be proving the following by filling in the blanks: If the set of coordinate vectors {[u,la u, a is linearly dependent in R", then the subset {u,u,} is linearly dependent in V. You need only write the word for each blank on our quiz, but be organized so I can grade your work. (a) If the set of vectors {u,a ·[u,] is linearly dependent in (b) then there exist scalars c,, (where at least one c, is non-zero), (c) such that = 0 the zero vector in R" . (d) By the of the coordinate mapping: G[u,]a+ +Cpup си. ...+C u + + + -.. (e) (Note: c,u, is a vector in (f) Весause 0. R" =|0, la, we can see that from above and part (c) that we have [qu, +.+c,u,]=[0, la: Initial if you agree +c̟u, pp ]B
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