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Vector Space (Vector Space (So far we've been looking at system of…

- - - - Surprisingly the answer is yes but only if a few conditions are met. Primarily if addition and multiplication are preserved
        
        We know addition and multiplication is preserved if all these axioms are met
      - Preservation of Addition
        
        This means the way we've been adding vectors should be the same for this new "space"
        
        Addition in this case means when we add an object they have the same number of entries and adding them means we're adding the respective elements
        
        So if we had S which stands for some generic space and two elements of S, \(y_k\) and \(z_k\) then \(y_k\) + \(z_k\)= \(y_{-2}+z_{-2},y+z..y_2+z_2\)
        
        Again we're just adding the respective elements of the objects together
      - Preservation of Multiplication
        
        This means like with vectors that every entry is multiplied by some scalar
        
        So 2\(y_k=2y_{-2},..2y,..2y_2,...\)
  - - - Can we treat \(P_n\) as a vector? In other words is \(P_n\) a vector space?
        
        \(P_n\) in this case applies only to \(n\geq 0\)
        p(t)=\(a_0+a_1t+a_2t^2+...+a_nt^n\)
        
        Is addition preserved?
        
        (p+q)(t)=p(t)+q(t)
        =\(a_0+b_0)+(a_1+b_1)t+...+(a_n+b_n)t^n\)
        
        Yes, as we can see adding two functions of polynomials is the same as adding the respective entries
        
        q(t)=\(b_0+b_1t+b_2t^2+...+b_nt^n\)
        
        Is multiplication preserved?
        
        (cp)(t)=cp(t)=\(ca_0+(ca_1)t+...+(ca_n)t^n\)
        
        Yes, every entry is multiplied by the scalar
  - - - This means that we can treat something really complicated as a simple vector and we have all the tools that we use on vectors available to use on that more complicated "space"
- - - - The issue is that when we're working with a smaller section of a known vector space we can into an issue where a few of the properties of vectors no longer apply
        
        This refers to 3 particular axioms 1,4,and 6. These were the zero vector is included and addition and multiplication is closed
        
        If the zero vector was not in our subset then it would make no sense if we added u with 0
        
        Closed under vector addition means that the sum of any two members also belongs to the set
        
        An example would be if our set is any even integer. If we add two even integers the result is an even integer
        
        Closed under (scalar) multiplication means that if we multiply a scalar with member the result is inside the set
        
        If we multiply any scalar to an even number we always get an even number so this would be an example of closed under multiplication
- - - - So column space is viewing the columns as vectors. Their span would be the column "space"
        
        So the column space is the same as finding the pivot columns of a matrix