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Using isomorphisms to transform polynomials to vectors.

  1. Nov 29, 2011 #1
    Does this actually work well? We won't learn isomorphisms in linear algebra, but a friend of mine showed me an example as I prefer to work with vectors and matrices rather than polynomials (All of my problem sets are with matrices and vectors).

    For example, if I wanted to find a basis for P3 that contains the polynomial 8x^3 - 2x^2 + 5x + 11, could you use isomorphisms to transform it into a vector in R4 and then find a basis?
     
  2. jcsd
  3. Nov 29, 2011 #2
    Yes, sure. This can indeed be done.

    Send

    [tex]f(ax^3+bx^2+cx+d)=(a,b,c,d)[/tex]

    This can be shown to be an isomorphism. So the vector spaces [itex]P_3[/itex] and [itex]\mathbb{R}^4[/itex] are the same for all linear algebra purposes. So a basis with the polynomials can be found by searching a basis in [itex]\mathbb{R}^4[/itex] first.
     
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