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Linear Algebra! (Matrices (Row/Column space A = mxn matrix row space…

- - - - The columns of reduced echelon form satisfy same relations as original columns. Thus, we can remove vectors corresponding to free variables in reduced echelon form and we are left with a linearly indep. subset w/ same span
      - Pivot columns of A are those corresponding to leading terms of reduced echelon form
    - - nonzero rows of E form a basis for RowSpace(A) and rowRank(A) = number of nonzero rows in E
        
        RowRank(A) = ColRank(A)
        
        Rank
        Rank(A) = Row(rank(A) = Col(Rank(A)
  - - - If E is the reduced echelon form of A, then RowSpace(A) = RowSpace(E)
    - - Row operations don't change relations between columns
  - - - Pivot columns of A form a basis for ColSpace(A) and colRank(A) is the number of pivot columns
- - - - If S in T then span(S) in span(T)
      - S in V
        then span(S) is a subspace of V
      - S in V, v in V
        span(S and {v}) = span(S) iff v in span(S)
    - - S in V is linearly dependent iff there exists v in S s.t. v is in span(S-{v})
        
        Any finite S in V has a linearly independent subset w/ same span
      - Basis
        B in V is a basis if it is linearly indep. and span(B)=V
        An ordered basis is ordered in sequence
        
        B is a basis for V, every v in V can be expressed as a unique linear comb. of elements in B
        
        B = <v1, ... , vn>. Given v in V, there exists unique c1, ... , cn s.t. v = c1v1 + ... + cnvn. The coordinates of v w.r.t. B is [v]B = (c1 c2 ... cn) in F^n.
        
        coordinates define a function V --> F^n with v --> [v]B. It is bijective and preserves linearity.
        
        V is finite dimensional if it has a basis with finitely many elements.
        
        Exchange Lemma
        Can exchange vectors in a basis with other vectors and you still have a basis (see notes for exact statement)
        
        If V is finite dimensional, any two bases for B have same number of elements
        
        Dimension
        V is finite dim'l. Dim(V) = number of elements in any basis for V
        
        Dim(V) = n
        if S in V is linearly indep., then S has at most n elements
        
        Dim(V) = n:
        if S in V is linearly indep. it can be completed to a basis for V
        
        dim(V) = n
        if S in V has exactly n elements, then S is linearly indep. iff S spans V
        
        Once you have right number of elements, only need to check one of two conditions on being a basis.
        
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  - - - Range of f: V-->W linear transf. is the image of V under f (subspace of W). Rank of f is the dimension of R(f)
- - - - A general system is consistent iff vector of solutions are in the column space. The solution is unique iff n = rank(A)