) Suppose L : V → W is a linear transformation such that the kernel of L is trivial, that is, ker(L) = {0}. Prove that L is one-to-one. (use the definitions) Hint: Start with L(v1) − L(v2) = 0 and use linearity of L. (b) Prove the converse of (a), that is, suppose L : V → W is a one-to-one linear transformation and prove that ker(L) = {0}. (use the definitions) Hint: Start with v ∈ ker(L) and use the
) Suppose L : V → W is a linear transformation such that the kernel of L is trivial, that is, ker(L) = {0}. Prove that L is one-to-one. (use the definitions) Hint: Start with L(v1) − L(v2) = 0 and use linearity of L. (b) Prove the converse of (a), that is, suppose L : V → W is a one-to-one linear transformation and prove that ker(L) = {0}. (use the definitions) Hint: Start with v ∈ ker(L) and use the
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
(a) Suppose L : V → W is a linear transformation such that the kernel of L is trivial, that is,
ker(L) = {0}. Prove that L is one-to-one. (use the definitions)
Hint: Start with L(v1) − L(v2) = 0 and use linearity of L.
(b) Prove the converse of (a), that is, suppose L : V → W is a one-to-one linear transformation
and prove that ker(L) = {0}. (use the definitions)
Hint: Start with v ∈ ker(L) and use the definition of one-to-one to show that v = 0.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 3 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,
