Question

Let T be an element of L(R³) such that-4,5,$7$ are it eigenvalues. show that T(x)-9x=(-4,5,$7$ )

Answer 1

Since T has at most 3 distinct eigenvalues.

The hypothesis implies that 9 is not an eigenvalue of T.

Thus $T - 9I$ is surjective. In particular, there exists $x \space \epsilon \space R^3$ such that $(T − 9I)x = Tx − 9x = (-4,5, 7)$

The entries of this particular vector are a red herring: we could just as easily find a y exists $y \space \epsilon \space R^3$ such that $T y - 9y = (86,75,309)$ by the same argument.