PS1(2) If A, B = Mn(F) and a, BE F, prove that tr (aA + BB) = a tr A + 3 tr B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please follow the promot and use the facts provided only to prove the problem. Answer in latex please.
Transcribed Image Text:PS1(2) If A, B ≤ M₁ (F) and a, ß € F, prove that tr (aA + ßB) = a tr A + ß tr B.
![In this problem-set, F stands for C or R.
In this problem-set, you are allowed to use the following facts:
F1. If M € Mmxn(C), then the conjugate-transpose of M, denoted by M* = [m²¡], is the n-by-m
matrix such that mij = mji.
F2. If A € Mmxn (F) and B € Mnxp(F), then
F3. If 2₁,..., Zn € C, then
F4. If 21,..., Zn € C, then
[AB]ij
n
n
k=1
Σaikbkj.
k=1
n
24-Ź²
Zi =
k=1
Σzi.
n
n
II ²i = II Zi.
k=1
k=1
F5. If A ≤ M₁ (F), then the trace of A, denoted by tr A = tr A, is defined by tr A = Σk_1 akk.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7a75c22-10f0-489b-b950-0e4b5d1d948e%2Fad0e618c-8782-4718-b6b7-d6ec2cf1dba3%2F1gkz36_processed.png&w=3840&q=75)
Transcribed Image Text:In this problem-set, F stands for C or R.
In this problem-set, you are allowed to use the following facts:
F1. If M € Mmxn(C), then the conjugate-transpose of M, denoted by M* = [m²¡], is the n-by-m
matrix such that mij = mji.
F2. If A € Mmxn (F) and B € Mnxp(F), then
F3. If 2₁,..., Zn € C, then
F4. If 21,..., Zn € C, then
[AB]ij
n
n
k=1
Σaikbkj.
k=1
n
24-Ź²
Zi =
k=1
Σzi.
n
n
II ²i = II Zi.
k=1
k=1
F5. If A ≤ M₁ (F), then the trace of A, denoted by tr A = tr A, is defined by tr A = Σk_1 akk.
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