4.30 (Refer to Example 4.19. This exercise shows that the requirement that pairwise intersections be trivial is genuinely stronger than the requirement only that the intersection of all of the subspaces be trivial.) Give a vector space and three subspaces W₁, W2, and W3 such that the space is the sum of the subspaces, the intersection of all three subspaces W₁n W₂ W3 is trivial, but the pairwise intersections W₁ W₂, W₁ W3, and W₂ W3 are nontrivial.
4.30 (Refer to Example 4.19. This exercise shows that the requirement that pairwise intersections be trivial is genuinely stronger than the requirement only that the intersection of all of the subspaces be trivial.) Give a vector space and three subspaces W₁, W2, and W3 such that the space is the sum of the subspaces, the intersection of all three subspaces W₁n W₂ W3 is trivial, but the pairwise intersections W₁ W₂, W₁ W3, and W₂ W3 are nontrivial.
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
Please do Exercise 4.30 and please show step by step and explain
Transcribed Image Text:4.30 (Refer to Example 4.19. This exercise shows that the requirement that pairwise
intersections be trivial is genuinely stronger than the requirement only that the
intersection of all of the subspaces be trivial.) Give a vector space and three
subspaces W₁, W2, and W3 such that the space is the sum of the subspaces,
the intersection of all three subspaces W₁n W₂ W3 is trivial, but the pairwise
intersections W₁ W₂, W₁ W3, and W₂ W3 are nontrivial.
Transcribed Image Text:4.19 Example If there are more than two subspaces then having a trivial inter-
section is not enough to guarantee unique decomposition (i.e., is not enough to
ensure that the spaces are independent). In R³, let W₁ be the x-axis, let W₂ be
the y-axis, and let W3 be this.
W3q|q, r = R}
(O)
The check that R³ =W₁ + W₂+W3 is easy. The intersection W₁ W₂n W3 is
trivial, but decompositions aren't unique.
=
=
0-0----0-(7)-0-0)
+
=
+
+
+
(This example also shows that this requirement is also not enough: that all
pairwise intersections of the subspaces be trivial. See Exercise 30.)
Expert Solution
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
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
