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Let $T: P_3 \to P_3$ be defined by $$T(ax^3 + bx^2 + cx + d) = (a+b)x^3 + (b-a)x^2 + (a+b+d)x +(a-b+2c+d)$$

In lead-up problems, I've shown (1) that $P_1$ and $\mathcal{V} = \{p(x) : p(1) = 0 \}$ are both $T$-invariant subspaces of $P_3$, and (2) that $\mathcal{V}$ contains a 2-dimensional $T$-invariant subspace $\mathcal{W}$ for which $P_3 = P_1 \oplus \mathcal{W}$.

(I did (2) by showing a generic $P_1 = cx + d$ and a generic $w \in \mathcal{W} \ni w = ax^3 + bx^2$ both have 2-dimensional $T$-invariant subspaces -- i.e. $T$ of these generics gives us something independent but still in the subspace (so $T(w) \subseteq W$), while $T^2$ leads to a dependence relation.)

But lastly I'm supposed to use these findings to construct a basis $\mathcal{B}$ for which $[T]_{\mathcal{B}}$ is block diagonal, and I'm just not seeing it.

dj4
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    my impression is that you have not properly typed the result of the mapping. It has no $x^3$ term and two separate constant terms, in parentheses – Will Jagy May 01 '19 at 02:44
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    anyway, what you do is correctly construct the matrix $M$ with regards to the ordered basis $x^3,x^2,x,1.$ If not what you want, then find invertible $P$ such that $P^{-1}MP$ satisfies – Will Jagy May 01 '19 at 02:46
  • @Will, that method certainly works, but it doesn't use the $T$-invariant subspaces 636164 has already found. There should be a way to make use of that information. – Gerry Myerson May 01 '19 at 03:20
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    @Gerry I see what you mean. – Will Jagy May 01 '19 at 03:24
  • By “invariant subspace” do you mean $W$ such that $T(W)\subseteq W$ or $T(W)=W$? – amd May 01 '19 at 06:21
  • @WillJagy Yes, I did type it incorrectly, thanks. Edited. – dj4 May 01 '19 at 14:07
  • @amd I think we need $T(W) \subseteq W$. Edited. – dj4 May 01 '19 at 14:07
  • Found this (https://math.stackexchange.com/questions/383970/proving-a-diagonal-matrix-exists-for-linear-operators-with-complemented-invarian?rq=1) which seems relevant. – dj4 May 01 '19 at 14:14
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    If $\mathcal B$ is the union of bases for $P_1$ and $\mathcal W$, ordered appropriately, $[T]_{\mathcal B}$ will be block diagonal. – amd May 01 '19 at 19:41

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