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Euclidean Vector Space
In general, euclidean space is defined by any
is Euclidean length of a 2D vector.
Multiply all components by scaling factor
- stays parallel
- → result vector is same direction as original
- → result vector is in opposite direction.
Vector Addition and Subtraction
Apply operation to corresponding components
represents space of matrices
Let be a set with the operation of addition and scalar multiplication:
- (commutativity of addition)
- (associativity of addition)
- (additive identity)
- (additive inverse; denoted )
- (distributive 1)
- (distributive 2)
- (associativity of multiplication)
- (multiplicative identity)
represents the line .
is not space since it does not satisfy the closure properties: let ;
represents the space of continuous functions on the interval . All axioms and closure properties are satsified for this space.
Focus on for .
degree of (denoted ) is (max power of )
Let represent all polynomials of degree < . satisfies all properties and axioms and is therefore spaaaaaace!
is vector space, , then
- (additive inverse is unique)
For a space , is a subspace of if it also satisfies the closure properties
is a subspace of since it satisfies the closure properties