| Definition of vector space |
- Informally, a collection of vectors that can be added together and scaled by a scalar
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- Vector space
 over field  defines two operations
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- Vector addition:
 , denoted as 
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- Scalar multiplication:
 , denoted as 
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- If scalars are real numbers, then is called a real vector space
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- 8 properties of addition and multiplication operations:
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- Commutativity of vector addition:
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- Associativity of vector addition:
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- Additive identity:
 (zero vector), s.t. 
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- Additive inverse:
 , s.t. 
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- Multiplicative identity:
 (multiplicative identity of F), s.t. 
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- Associativity of scalar multiplication:
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- Distributivity of scalar sums:
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- Distributivity of vector sums:
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