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I don't know how to solve this problem - I have to transform the coordinates of a bilinear form that has matrix $$ \begin{pmatrix} 1 & 2 & -2 \\ 2 & -2 & 3 \\ -2 & 3 & 2 \\ \end{pmatrix} $$ in base $$ \begin{pmatrix} -3 & 1 & 0 \\ 5 & -2 & 3 \\ 2 & 0 & -1 \\ \end{pmatrix} $$ To matrix of the same bilinear form in the base $$ \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \\ \end{pmatrix} $$ How do I do it? Should there be scalar products of the base vectors in the matrix(but then I do not understand what is there the first base for) or can I just find a transition matrix from one base to the other and just multipy it?

Thank you.

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Yes, you need the change of basis matrix $P$ whose columns are the coefficients of the new basis vectors when you express them as linear combinations of the old. I'll let you figure out how to get that from the two matrices directly. Then the expression for the bilinear form in the new basis should be $P^TAP$, where its matrix representation in the old basis is $A$. (To convince yourself of this, remember that the coordinate vector of a vector $x$ with respect to the new basis will be $x'=Px$. So $x^TAy = (Px')^TA'(Py') = \dots$.)

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