a) The proof is by contradiction. So we begin by supposing that z + <1> _ and w # _< 2 > _ (which is the negation of what we're trying to prove). b) Since z # < 3> , it follows that z has an inverse z < 4> . -1 such that z-1.z : c) Since z · w = 0, we can multiply both sides of this equation by < 5 > and obtain the equation w = <6> . This equation contradicts the supposition that _<7> . d) Since our supposition has led to a false conclusion, it follows that our supposition must be_< 8 > _. Therefore it cannot be true that_<9> _, so it must be true that_< 10 >
a) The proof is by contradiction. So we begin by supposing that z + <1> _ and w # _< 2 > _ (which is the negation of what we're trying to prove). b) Since z # < 3> , it follows that z has an inverse z < 4> . -1 such that z-1.z : c) Since z · w = 0, we can multiply both sides of this equation by < 5 > and obtain the equation w = <6> . This equation contradicts the supposition that _<7> . d) Since our supposition has led to a false conclusion, it follows that our supposition must be_< 8 > _. Therefore it cannot be true that_<9> _, so it must be true that_< 10 >
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 show each step
Transcribed Image Text:(a) The proof is by contradiction. So we begin by supposing that z
<1> _ and uw + < 2 >_ (which is the negation of what we're trying
to prove).
+
(b) Since z + _< 3 > _, it follows that z has an inverse z-1 such that z-1.z
<4> .
•2 =
(c) Since z· w = 0, we can multiply both sides of this equation by_ < 5 >
and obtain the equation w =
supposition that <7>
%3D
< 6 > . This equation contradicts the
(d) Since our supposition has led to a false conclusion, it follows that our
supposition must be < 8 >_. Therefore it cannot be true that_< 9 > _,
so it must be true that < 10 > .
Transcribed Image Text:Proposition 4.2.11. Given that z = a + bi, w = c+ di, and z· w = 0.
Then it must be true that either z = 0 or w = 0.
The proof of Proposition 4.2.11 is outlined in the following exercise.
Exercise 4.2.12. Complete the proof of Proposition 4.2.11 by filling in the
blanks. Note that some blanks may require an expression, and not just a
single number or variable.
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