Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let L(R") be the set of linear transformations T : R" →→ R² with the metric ||T – ||. Recall from Linear Algebra that the following are equivalent for T E L(R"): T is invertible + T is injective + T is surjective + det(AT) # 0, where ¿(T,S) AT is the matrix of T. Since det is a continuous function, it follows that the set GL, of invertible linear transformations is open in L(R"). Prove the following more precise statement: Let T E GLn and let r = ||T-1"'. Then the r-neighborhood of T in L(R") is contained in GL,n. Hint: Show that for any S E L(R") and v E R" we have r• lv| < ||T – S|| · |v| + |S(v)|.

Let L(R") be the set of linear transformations T : R" →→ R² with the metric ||T – ||. Recall from Linear Algebra that the following are equivalent for T E L(R"): T is invertible + T is injective + T is surjective + det(AT) # 0, where ¿(T,S) AT is the matrix of T. Since det is a continuous function, it follows that the set GL, of invertible linear transformations is open in L(R"). Prove the following more precise statement: Let T E GLn and let r = ||T-1"'. Then the r-neighborhood of T in L(R") is contained in GL,n. Hint: Show that for any S E L(R") and v E R" we have r• lv| < ||T – S|| · |v| + |S(v)|.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please solve as soon as possible

Let L(R") be the set of linear transformations T : R" → R" with the metric d(T,S) = ||T – ||. Recall from Linear Algebra that the following are equivalent for T E L(R"): T is invertible + T is injective T is surjective + det(AT) 0, where AT is the matrix of T. Since det is a continuous function, it follows that the set GLn of invertible linear transformations is open in L(R"). Prove the following more precise statement: Let T E GLn and let r = ||T-1||. Then the r-neighborhood of T in L(R") is contained in GLµ. Hint: Show that for any S E L(R") and v E R" we have r. |미 < ||T-SI| •미 + \S(») .
Transcribed Image Text:Let L(R") be the set of linear transformations T : R" → R" with the metric d(T,S) = ||T – ||. Recall from Linear Algebra that the following are equivalent for T E L(R"): T is invertible + T is injective T is surjective + det(AT) 0, where AT is the matrix of T. Since det is a continuous function, it follows that the set GLn of invertible linear transformations is open in L(R"). Prove the following more precise statement: Let T E GLn and let r = ||T-1||. Then the r-neighborhood of T in L(R") is contained in GLµ. Hint: Show that for any S E L(R") and v E R" we have r. |미 < ||T-SI| •미 + \S(») .
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
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,
SEE MORE TEXTBOOKS