linear combos everywhere

• a linear combination is a finite sum of the form
  
  • [Shankar] calls it a superposition. ([Shankar] §1.2 p. 9).
    
    
    
    • there is also something called the superposition principle:
      
      
      
      • F(x)=A,F(y)=B ⇒ F(x+y)=A+B
        
        
      
      
      • additivity property of functions making them linear functions,aka lin. operators and lin. maps.
        
        
      
      
      
      
    
    
    
    
  
  
  • a finite sum of that form
    
    ⇔ summation of terms
    
    ⇔ sum of products
    
    
  
  
  
  


• One clear physical meaning of the linear combination is that it is a weighted sum.
  

• instances of linear combination:
  
  
  
  
  
  • linear combinations are thus a key primitive in - mathematically speaking - "nature".
    
    
  
  
  • instances of linear combination in inner products or related by the inner product:
    
    
    
    
    • inner product is a class of operations having the same set of properties,e.g.,
      
      
    
    
    
    • vector dot product
      
      
      
      
      • of two vectors A⋅B=Σaibi , and in its variant, matrix multiplication , thus
        
        
      
      
      
      
    
    
    • matrix multiplication
      
      
      
      
      • , for each row in the mxn premultiplier elements from each column in the row are linearly combined (linearly combine) with elements from the corresponding row element in the postmultiplier giving a single element in the product matrix
        
        
        
      
      
      • cij = ai1b1j + ai2b2j + ai3b3j + … + ainbnj
        
        
      
      
      
      
    
    
    • integration operator
      
      
      
      
      • 1.bis2- in the approximation of integration by finite sums
        
        
      
      
      
      • Σf(ci)Δx , where ci is a point in the ith subinterval
        
        
      
      
      • the integral itself expressing a wide class of functions eg, area,distance,path length,volume,work,flow,etc. = ∫f=limΣf(x)⋅(Δx→0)
        
        
      
      
      • n.b. the integrat operator is a linear transformation, which have linear combination as a property, see below.
        
        
      
      
      
      
    
    
    • N.B. N.B. the equivalence of integrals and the dot product. Explicit examples:
      
      
      
      
      
      • [Waerden]§1: where to express that state functions Ψ are Lebesgue-integrable, he writes <Ψ,Ψ> = ∫ΨΨ*dq, where I take the LHS to be an inner product.
        
        
      
      
      • in investigating, we can take this parallelism further by emphasizing the coef:
        
        
        
        
        • case: dot product, matrix multiplication and determinant: coefs is a "vector" component
          
          
          
        
        
        • case: finite sum: coef. is subinterval width, Δx=b-a/n and kth x is x_k=x0+k(b-a/n)
          
          
        
        
        
        
      
      
      
      
    
    
    • The Unit form , Hermitean product <u,v>
      
      
      
      
      
      • the unit form <v,v> = ∑ c_k^* c_k ([Waerden] §9)
        
        
      
      
      • is pretty damn close to the way
        
        
        
        
        • state functions Ψn of the Schrodinger equation are "integrable in the sense of lebesgue" , ie <Ψ,Ψ>=∫Ψ^*Ψ ([Waerden] §1)
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Determinant calculation , for orders 2 and 3 at least ?
    
    
  
  
  • the definition of a vector itself. As a vector is expressible as a linear combination of the magnitudes of its components and the orthonormal basis vectors, better known to us as unit coordinate vectors.
    
    
    
    
    • eg, a vector in Euclidian space may be expressed as
      
      
      V = v_x(1,0,0)+v_y(0,1,0)+v_z(0,0,1) = +v_x.i++v_y.j+v_z.k .
      
      see theorem 12.6 in [Aposotl I].
      
      
    
    
    • a vector in vector space of n-tuples V_n is expressed as a linear combination of the space's unit coordinate vectors (eg, i,j,k for V₃).
      
      
      
      
      
      • [ApostolI]ch15,§15.6: definition: the set of all fin. linear comb. of elmts of linear space S
        
        
        
        
        • satisfies closure,
          
          
        
        
        • is a subspc of S and
          
          
        
        
        • is called the span of S, or subspc spanned by S.
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Linear transformation: application of Linear transformation A to finite-dimensional vector space V:
    
    
    
    
    • lin. xforms are matrix multiplications, so by extension of this and by definition ipso facto are also linear combinations
      
      
    
    
    • 
      
      
      
    
    
    • !@ see third property of linear transformations in [Apostol II] 2.1. !@
      
      
    
    
    
    
  
  
  • Gram-schmidt process of orthogonalization
    
    
    
    
    • (mapping among either basis sets or axes or both)
      
      
      
    
    
    • [Apostol II] pp.24-26.
      
      
    
    
    
    
  
  
  
  

• The definition of a linear manifold [von Neumann] §II.1 p. 38.




• "a subset U of a linear space R is called a linear manifold if it contains all linear combinations
  
  a₁f₁ + … + a_kf_k for any k of elements f₁, ... , f_k.




• "[sufficiently requiring]" ( f,g ∈ R ) ⇒ ( af, f+g ∈ R )




• .'. f₁,…,f_k ∈ U ⇒ a₁f₁ , a₁f₁+a₂f₂, a₁f₁+ a₂f₂+a₃f₃, ..., a₁f₁+...+a_kf_k ∈ U


• or, put slightly differently,


• .'. ∀ f₁,...,f_k ∈ U   a₁f₁ , a₁f₁+ a₂f₂, a₁f₁+ a₂f₂+a₃f₃, …, a₁f₁+...+a_kf_k ∈ U.






• U is ⊆ R ⇒ the set a₁f₁ + … + a_kf_k ∀ k=1,2,…, a₁,…,a_k in ℂ, f₁, …, f_k in U , it is a subset of every other linear manifold , which is then said to be spanned by U.