Matrix is $$ \begin{pmatrix} x^3 & x & 1 \\ x^3 & -x^2 & 1 \\ x^3 & kx^2 & kx \end{pmatrix} $$ I reduced it down to $$ \begin{pmatrix} 0 & x + x^2 & 0 \\ x^3 & -x^2 & 1 \\ 0 & k & kx - 1 \end{pmatrix}, $$ but I'm stuck at this point. Any advice or hints?
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$\begingroup$ Your question is unclear: What is $P$ and why do some $x$ have indices and some not? $\endgroup$ – Ramanujan Feb 8 '19 at 16:50
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$\begingroup$ @ViktorGlombik the numbers following the $x$'s are likely exponents. $\endgroup$ – Dave Feb 8 '19 at 16:53
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$\begingroup$ @Dave I edited to reflect that, but there are some $x_2$s and $x_3$s and some $x$s without index. $\endgroup$ – Ramanujan Feb 8 '19 at 16:56
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$\begingroup$ sorry i didn't finish writing the question. P should be P3(R). $\endgroup$ – Kayy Wang Feb 8 '19 at 17:03
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$\begingroup$ Which are the polynomial with real coefficients and have degree 3 or less? $\endgroup$ – Ramanujan Feb 8 '19 at 17:04
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Think of the basis vectors $x, x^2, x^3, 1$ then the matrix is $$ \begin{pmatrix} 1 & 0 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ 1 & k & k & 0 \end{pmatrix} $$ Then find the $k$ which do not make its rows linearly dependent.
