(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that X + Y is a direct sum if and only if X ^ Y = {0}.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted XY) if every
ve X + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that
X + Y is a direct sum if and only if X ^ Y = {0}.
Transcribed Image Text:(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted XY) if every ve X + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that X + Y is a direct sum if and only if X ^ Y = {0}.
Expert Solution
Step 1: Symbolic notation

Advanced Math homework question answer, step 1, image 1

steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,