Suppose {v₁, ..., vr} is linearly independent but does not generate the vector space V. Show there exists a vector Vr+1 such that {V₁,..., Vr, Vr+1} is linearly independent. Hint: Since the set is not generating, there is a vector Vr+1 that is not a linear combination of the vį (i = 1, ..., r). Then show that (v₁,..., Vr+1} is linearly independent.
Suppose {v₁, ..., vr} is linearly independent but does not generate the vector space V. Show there exists a vector Vr+1 such that {V₁,..., Vr, Vr+1} is linearly independent. Hint: Since the set is not generating, there is a vector Vr+1 that is not a linear combination of the vį (i = 1, ..., r). Then show that (v₁,..., Vr+1} is linearly independent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Suppose {v₁,..., Vr} is linearly independent but does not generate the vector
space V. Show there exists a vector Vr+1 such that {V₁,..., Vr, Vr+1} is linearly independent.
Hint: Since the set is not generating, there is a vector Vr+1 that is not a linear combination
of the vį (i = 1,...,r). Then show that {v₁,..., Vr+1} is linearly independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F818163d5-e058-4810-8470-4c2365a08cf2%2Fy824bbw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose {v₁,..., Vr} is linearly independent but does not generate the vector
space V. Show there exists a vector Vr+1 such that {V₁,..., Vr, Vr+1} is linearly independent.
Hint: Since the set is not generating, there is a vector Vr+1 that is not a linear combination
of the vį (i = 1,...,r). Then show that {v₁,..., Vr+1} is linearly independent.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)