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Let V1, V2, V3 be a basis of a vector space V over R. Is V1 + V2, V2 + 2v3, v3 + 3v₁ also a basis? What about over the field F7 of 7 elements?

Let V1, V2, V3 be a basis of a vector space V over R. Is V1 + V2, V2 + 2v3, v3 + 3v₁ also a basis? What about over the field F7 of 7 elements?

Advanced Engineering Mathematics
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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Let V1, V2, V3 be a basis of a vector space V over R. Is v₁ + V2, V2 + 2v3, v3 + 3v₁ also a basis? What about over the field F7 of 7 elements? Select one: Yes over R, no over F7 Yes for both fields O No over R, yes over F7 O This depends on the specific v₁, V2, V3 as elements of V No for both fields
Transcribed Image Text:Let V1, V2, V3 be a basis of a vector space V over R. Is v₁ + V2, V2 + 2v3, v3 + 3v₁ also a basis? What about over the field F7 of 7 elements? Select one: Yes over R, no over F7 Yes for both fields O No over R, yes over F7 O This depends on the specific v₁, V2, V3 as elements of V No for both fields
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Step 1: Solution

Given: Let v subscript 1 comma space v subscript 2 and v subscript 3 be a basis of vector space V over straight real numbers

We have to check  v subscript 1 plus v subscript 2 comma end subscript space v subscript 2 plus 2 v subscript 3 comma space end subscript v subscript 3 plus 3 v subscript 1 space is also a basis or not.

and also what about over the field double-struck F subscript 7 of 7 elements ?

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