Let V denote a vector space over F. Suppose {B1, B2,..., B1, B41} is linearly independent in V and {B1, B2,..., B1, B41, a+1, Q+2, ..., an} is linearly dependent in V. Show that one of the a; is dependent on the remaining vectors. To do this, assume Ci B1 + c2 B2 + ·..+ cißi + C41B+1 + d41Q+1 + di+20+2 + ...+ dnan = and show that one of di41, di42,..., dn must be nonzero.
Let V denote a vector space over F. Suppose {B1, B2,..., B1, B41} is linearly independent in V and {B1, B2,..., B1, B41, a+1, Q+2, ..., an} is linearly dependent in V. Show that one of the a; is dependent on the remaining vectors. To do this, assume Ci B1 + c2 B2 + ·..+ cißi + C41B+1 + d41Q+1 + di+20+2 + ...+ dnan = and show that one of di41, di42,..., dn must be nonzero.
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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Transcribed Image Text:Let V denote a vector space over F.
Suppose {B1, B2,..., B1, B1+1} is linearly independent in V and
{B1, B2, ..., B1, Bi41, a+1, QL+2,... , an} is linearly dependent in V.
Show that one of the a; is dependent on the remaining vectors.
To do this, assume
ci B1 + c2 B2 + ·..+ cißı + C41Bi+1 + d+1¤+1 + di+2¤+2 + •.+ dnɑn =
and show that one of di41, d1+2,..., dn must be nonzero.
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