Problem 1. Let K be a field. Show that every finitely generated K-module is a finite-dimensional K-vector space. (Hint: let V be a vector space such that v₁,...,Um generate V as a K-module for some m € Z₂1- Show that dim(V) < m.)
Problem 1. Let K be a field. Show that every finitely generated K-module is a finite-dimensional K-vector space. (Hint: let V be a vector space such that v₁,...,Um generate V as a K-module for some m € Z₂1- Show that dim(V) < m.)
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
![Problem 1. Let K be a field. Show that every finitely generated K-module is a finite-dimensional K-vector
space. (Hint: let V be a vector space such that v₁,...,Um generate V as a K-module for some m € Z>₁.
Show that dimK (V) < m.)
Problem 2. (a) Show that {1} is a Q[t]-basis of Q[t].
(b) Show that {1, t, t², ³,...} is a Q-basis of Q[t].
Problem 3. Let f(t) = K[t] be any polynomial of degree n > 1. Show that I, t, f,...,1 form a K-basis
for K[t]/(f(t)).
Problem 4 (Math 121A / Math 3A). Consider
Y =
1₁
which we consider as a matrix over Q.
(a) Compute the eigenvalues of Y. What are their eigenvectors? Prove what you claim.
(b) Find an explicit 2 x 2 invertible matrix P over Q such that P-¹YP is a diagonal matrix.
Problem 5 Let K be a field and U, V be finitely-dimensional K-vector spaces (or K-modules). Consider
any K-linear map : U → V.
(a) Show that for any K-submodule U' of U, we have dimk (U/U') = dimk (U) - dimk (U'). (Hint: Fix any
basis u₁,..., um of U'. By the basis extension theorem of Math 121A, you can find Um+1, um +2,..., Un E U
such that u₁, U2,..., un form a K-basis for U. Show that {um+1, um +2,...,un) is a K-basis for U/U'.)
(b) (Rank-Nullity theorem) Prove that dimk (U) = dimk (ker(p)) + dimk ((U)). (Hint: Use the fact that
U/ker (p) (U) as K-vector spaces.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e1260fe-93bd-4037-b0c5-e1f4a089cedf%2F50878e84-cca3-4850-88ac-04b1bea9c70a%2Fw1j3rk_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. Let K be a field. Show that every finitely generated K-module is a finite-dimensional K-vector
space. (Hint: let V be a vector space such that v₁,...,Um generate V as a K-module for some m € Z>₁.
Show that dimK (V) < m.)
Problem 2. (a) Show that {1} is a Q[t]-basis of Q[t].
(b) Show that {1, t, t², ³,...} is a Q-basis of Q[t].
Problem 3. Let f(t) = K[t] be any polynomial of degree n > 1. Show that I, t, f,...,1 form a K-basis
for K[t]/(f(t)).
Problem 4 (Math 121A / Math 3A). Consider
Y =
1₁
which we consider as a matrix over Q.
(a) Compute the eigenvalues of Y. What are their eigenvectors? Prove what you claim.
(b) Find an explicit 2 x 2 invertible matrix P over Q such that P-¹YP is a diagonal matrix.
Problem 5 Let K be a field and U, V be finitely-dimensional K-vector spaces (or K-modules). Consider
any K-linear map : U → V.
(a) Show that for any K-submodule U' of U, we have dimk (U/U') = dimk (U) - dimk (U'). (Hint: Fix any
basis u₁,..., um of U'. By the basis extension theorem of Math 121A, you can find Um+1, um +2,..., Un E U
such that u₁, U2,..., un form a K-basis for U. Show that {um+1, um +2,...,un) is a K-basis for U/U'.)
(b) (Rank-Nullity theorem) Prove that dimk (U) = dimk (ker(p)) + dimk ((U)). (Hint: Use the fact that
U/ker (p) (U) as K-vector spaces.)
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 3 steps with 51 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,
