Difference between revisions of "12-240/Classnotes for Thursday September 27"

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{{12-240/Navigation}}
 
{{12-240/Navigation}}
  
In this course, we will be focusing on both a practical side and a theoretical side.
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'''Vector Spaces'''
  
 
== Vector space axioms ==
 
== Vector space axioms ==
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''(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
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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
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== Theorems ==
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1.a x + z = y + z => x = y
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1.b ax = ay, a != 0, => x = y
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1.c ax = bx, x != 0, => a = b
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 +
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2. 0 is unique.
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3. Additive inverse is unique.
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4. 0_F * x = 0_V
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5. a * 0_V = 0_V
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 +
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6. (-a) x = -(ax) = a(-x)
 +
 +
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7. cx = 0 <=> c = 0 or x = 0_V

Revision as of 20:26, 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