I came across the following question:Let V be the real vector space of polynomials with degree ≤ 2. For α , β ∈ R let W α = { f ∈ V | f ( α ) = 0 } and D β = { f ∈ V | f ′ ( β ) = 0 }. f ′ describes the derivative of f.Find a basis for W α and D β .Unfortunatly I am not quite getting my head wrapped around vector spaces with polynomials yet. So after failing the question I had a look at the solution and there it says:By solving the system of linear equations: W α : { X − α , X 2 − α 2 } D β : { 1 , X 2 − 2 β X }More explanation is not given.I know that the elements in W α in the form of f ( x ) = c 2 X 2 + c 1 X + c 0 are zero at α, so f ( α ) = c 2 α 2 + c 1 α + c 0 = 0.Respectively for the elements in D β : f ′ ( β ) = 2 c 2 β + c 1 = 0.However, it is not clear to me how I get from that information to the basis of W α and D β using systems of linear equations.

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I came across the following question:Let   V be the real vector space of polynomials with degree   ≤  2. For   α  ,  β  ∈      R   let       W    α    =  {  f  ∈  V      |    f  (  α  )  =  0  } and       D    β    =  {  f  ∈  V      |        f    ′    (  β  )  =  0  }.       f    ′   describes the derivative of   f.Find a basis for       W    α   and       D    β  .Unfortunatly I am not quite getting my head wrapped around vector spaces with polynomials yet. So after failing the question I had a look at the solution and there it says:By solving the system of linear equations:      W    α    :  {  X  −  α  ,      X    2    −      α    2    }      D    β    :  {  1  ,      X    2    −  2  β  X  }More explanation is not given.I know that the elements in       W    α   in the form of   f  (  x  )  =      c    2        X    2    +      c    1    X  +      c    0   are zero at   α, so   f  (  α  )  =      c    2        α    2    +      c    1    α  +      c    0    =  0.Respectively for the elements in       D    β  :      f    ′    (  β  )  =  2      c    2    β  +      c    1    =  0.However, it is not clear to me how I get from that information to the basis of       W    α   and       D    β   using systems of linear equations.

jugf5 · Accepted Answer

Note that for   W form the condition  f  (  α  )  =      c    2        α    2    +      c    1    α  +      c    0    =  0    ⟹        c    0    =  −      c    2        α    2    −      c    1    α

I came across the following question: Let V be the real vector space of polynomials with deg

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