Using isomorphisms to transform polynomials to vectors.

[-Dragoon-]

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?

 

[micromass]

Yes, sure. This can indeed be done.

Send

$$f(ax^3+bx^2+cx+d)=(a,b,c,d)$$

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