The matrix whose columns are the coordinate vectors of the Hermite polynomials relative to the
standard basis {1,t,t²,t³} of P3 is given by:
[tex]\left[\begin{array}{cccc}1&0&-2&0\\0&2&0&-12\\0&0&4&0\\0&0&0&8\end{array}\right][/tex]
Since this matrix is already in row echelon form and there are 4 nonzero pivots, then its columns are linearly
independent. Since the coordinate vectors form a linearly independent set, then the Hermite polynomials
form a linearly independent set in P3: The dimension of P3 is 4; so this set of Hermite polynomials forms a basis for P3:
To learn more about Hermite polynomials from the given link
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