Difference between revisions of "12-240/Classnotes for Thursday September 27"
From Drorbn
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{{12-240/Navigation}} | {{12-240/Navigation}} | ||
− | + | '''Vector Spaces''' | |
== Vector space axioms == | == Vector space axioms == | ||
+ | |||
''(Quick recap)'' | ''(Quick recap)'' | ||
+ | |||
VS1. x + y = y + x | VS1. x + y = y + x | ||
+ | |||
VS2. (x + y) + z = x + (y + z) | VS2. (x + y) + z = x + (y + z) | ||
+ | |||
VS3. 0 vector | VS3. 0 vector | ||
+ | |||
VS4. + inverse -> - | VS4. + inverse -> - | ||
+ | |||
VS5. 1x = x | VS5. 1x = x | ||
+ | |||
VS6. a(bx) = (ab)x | VS6. a(bx) = (ab)x | ||
+ | |||
VS7. a(x + y) = ax + ay | VS7. a(x + y) = ax + ay | ||
+ | |||
VS8. (a+b)x = ax + bx | VS8. (a+b)x = ax + bx | ||
+ | |||
+ | == Theorems == | ||
+ | |||
+ | 1.a x + z = y + z => x = y | ||
+ | |||
+ | 1.b ax = ay, a != 0, => x = y | ||
+ | |||
+ | 1.c ax = bx, x != 0, => a = b | ||
+ | |||
+ | |||
+ | |||
+ | 2. 0 is unique. | ||
+ | |||
+ | |||
+ | 3. Additive inverse is unique. | ||
+ | |||
+ | |||
+ | 4. 0_F * x = 0_V | ||
+ | |||
+ | |||
+ | 5. a * 0_V = 0_V | ||
+ | |||
+ | |||
+ | 6. (-a) x = -(ax) = a(-x) | ||
+ | |||
+ | |||
+ | 7. cx = 0 <=> c = 0 or x = 0_V |
Revision as of 21:26, 27 September 2012
|
Vector Spaces
Vector space axioms
(Quick recap)
VS1. x + y = y + x
VS2. (x + y) + z = x + (y + z)
VS3. 0 vector
VS4. + inverse -> -
VS5. 1x = x
VS6. a(bx) = (ab)x
VS7. a(x + y) = ax + ay
VS8. (a+b)x = ax + bx
Theorems
1.a x + z = y + z => x = y
1.b ax = ay, a != 0, => x = y
1.c ax = bx, x != 0, => a = b
2. 0 is unique.
3. Additive inverse is unique.
4. 0_F * x = 0_V
5. a * 0_V = 0_V
6. (-a) x = -(ax) = a(-x)
7. cx = 0 <=> c = 0 or x = 0_V