Matrix Modern
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Matrix modern

Modern Abstract Algebra Proof - any help would be greatly appreciated?
Let H be the subset of M2(R) consisting of all matrices of the form [ a-b]
[b a]
for a,b belong to R
a.) show that
b.) show that
For ease, I'm going to denote that matrix [a b] instead of it's normal 2x2 representation.
You want to use the isomorphism:
φ( [a b] ) = a + b i
This is clearly a bijection. All you have to do is show that it is a homeomorphism, and to do that you need to prove:
Part a: φ( [a b] + [c d] ) = φ( [a b] ) + φ( [c d] )
Part b: φ( [a b] * [c d] ) = φ( [a b] ) φ( [c d] )
PART A:
This will be pretty trivial. First find [a b] + [c d]
[a -b] + [c -d] = [a+c -b-d ]
[b a] [d c] [b+d a+c ]
This means [a b] + [c d] = [a+c b+d].
And so φ( [a b] + [c d] ) = φ( [a+c b+d] ) = a+c+(b+d)i
Now look at φ( [a b] ) + φ( [c d])
= a + bi + c + di = a+c+(b+d)i
So we conclude φ( [a b] + [c d] ) = φ( [a b] ) + φ( [c d])
Thus this is a homeomorphism and since it's bijective, we know this is an isomorphism.
PART B:
First find [a b] * [c d]
[a -b] [c -d] = [ac-bd -bc-ad ]
[b a] [d c] [bc+ad ac-bd ]
Which means [a b] * [c d] = [ac-bd ad+bc]
Thus φ( [a b] * [c d] ) = ac-bd + (ad+bc) i
Now we look at φ( [a b] ) φ( [c d] )
= (a+bi)(c+di) = ac + bci + adi + bdi²
= ac + (bc+ad)i - bd = ac-bd + (ad+bc)i
Thus φ( [a b] * [c d] ) = φ( [a b] ) φ( [c d] )
And we know it's an isomorphism (bijective homeomorphism).
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![]() | MATRIX GAMES CLOSE COMBAT MODERN TACTICS - NEW RETAIL | ![]() | ![]() | US $39.99 | 20d 23h 48m |
![]() | Matrix Modern Black w/Glass 3Pc Coffee Table SET - HOUSTON ONLY! | ![]() | US $100.00 | 17h 8m |
![]() | NWT Mens Calvin Klein Layering T-Shirt T-Matrix Modern | ![]() | ![]() | US $12.99 | 29d 22h 54m |
![]() | SMA MATRIX Modern PLATFORM BED, Italy | ![]() | US $2,155.55 | 28d 22h 27m |
![]() | Modern Structural Analysis: The Matrix Method Approach by Anthony E. Armenakas | ![]() | ![]() | US $49.95 | 27d 4h 59m |
![]() | Bordering on the Body: The Racial Matrix of Modern Fict | ![]() | ![]() | US $9.95 | 25d 12h 30m |
![]() | Damping Matrix (1x) Mirrodin FOIL Modern Magic MTG *EliteMagicCards2* | ![]() | ![]() | US $5.95 | 25d 2h 53m |
![]() | New Modern Durable Area Rugs Carpet Matrix Gold 8x11 | ![]() | ![]() | US $592.00 | 17d 22h 35m |
![]() | Modern Matrix Algebra by Bernard Kolman and David R.... | ![]() | ![]() | US $14.95 | 15d 20h 7m |
![]() | Theory Construction and Selection in Modern Physics: The S Matrix James T. Cushi | ![]() | ![]() | US $237.49 | 14d 1h 53m |
![]() | Theory Construction and Selection in Modern Physics: The S Matrix James T. Cushi | ![]() | ![]() | US $112.62 | 13d 19h 10m |
![]() | Introduction to Modern Algebra and Matrix Theory Schrei | ![]() | ![]() | US $40.20 | 12d 22h 44m |
![]() | STAN GETZ modern world of LP 11 trk mono pressing black/blue label matrix a1n/b1 | ![]() | ![]() | US $26.03 | 12d 10h 56m |
![]() | Introduction to Modern Algebra and Matrix Theory NEW | ![]() | ![]() | US $22.50 | 11d 14h 27m |
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Modern Jazz: Matrix (Suite)











































Action films have become post-modern parodies of themselves. I wish I'd burned my DVD player after seeing the Matrix for the 1st time.
/ ITtelligenceDirAugust 24, 2010, 11:33 am