Let Rª be a vector space. Recall that u is a linear combination of vectors v1, ..., Um E R" if u = C1V1 +…+ CmVm Cor some c1, ..., Cm E R. (a) Let V = {v1, ..., Vm} C R". We want to recall a few definitions from linear algebra: i. Give the definition for span V, the span of V. ii. Give the definition for V to be linearly independent. iii. Give the definition for V to be a basis of R". (b) Let V1, V2 be different bases of R" and let v e Vị \ V2. Show that there must be a u E V½ \ Vị such that (Vị \ {v}) U {u} is a basis of R".
Let Rª be a vector space. Recall that u is a linear combination of vectors v1, ..., Um E R" if u = C1V1 +…+ CmVm Cor some c1, ..., Cm E R. (a) Let V = {v1, ..., Vm} C R". We want to recall a few definitions from linear algebra: i. Give the definition for span V, the span of V. ii. Give the definition for V to be linearly independent. iii. Give the definition for V to be a basis of R". (b) Let V1, V2 be different bases of R" and let v e Vị \ V2. Show that there must be a u E V½ \ Vị such that (Vị \ {v}) U {u} is a basis of R".
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 R" be a vector space. Recall that u is a linear combination of vectors v1, ..., vm E R" if
u = C1V1 + ……+ CmVm
for some c1, . .., Cm E R.
(a) Let V = {v1, . .. , Vm} Ç R". We want to recall a few definitions from linear algebra:
i. Give the definition for span V, the span of V.
ii. Give the definition for V to be linearly independent.
iii. Give the definition for V to be a basis of R".
(b) Let V1, V2 be different bases of R" and let v e Vị \ V2. Show that there must be a
u E V2 \ Vị such that (Vị \ {v}) U {u} is a basis of R".
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