Difference between revisions of "12240/Classnotes for Thursday September 27"
From Drorbn
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{{12240/Navigation}}  {{12240/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