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VectorSpace.md (896B)

      1 # Vector Space 
      2 
      3 **Source:** Linear Algebra Done Right
      4 
      5 **Chapter:** 1
      6 
      7 **Definition:** A vector space is a space where we find a closure under vector addition and scalar multiplication.
      8 
      9 Along with this, the following must be true:
     10 
     11 1. Commutative, a + b = b + a
     12 2. Associative, a(b * c) = b * (a * c) and a + (b + c) = b + (a + c)
     13 3. Additive Identity, a + 0 = a
     14 4. Additive Inverse, a + -a = 0
     15 5. Multiplicative Identity, 1a = a
     16 6. Distributive, a(u + v) = au + av and (a + b)u = au + bu
     17 
     18 ## Related Information
     19 
     20 When defining a vector space we define it as a set $V$ along with an addition and scalar multiplication on $V$ that satisfies the prior properties.
     21 
     22 We define addition and scalar multiplication as functions.
     23 
     24 The addition function can be:
     25 a : (V)^2 -> B : f(n,m) = n+m for all n,m in V.
     26 
     27 The multiplication function can be:
     28 m : (V,F) -> V : m(v,f) = vf for all v in V and c in F.