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### Prove that there are no bases in $\mathbb R^3$ in which $f$ has the matrix $B$

If $B$ and $B'$ are bases of $\Bbb R^3$, then what you did shows that the matrix $[f]_B^B$ of $f$ with respect to the basis $B$ cannot possible be equal to $[f]_{B'}^{B'}$. But the idea is to prove ...
1 vote

### Prove that there are no bases in $\mathbb R^3$ in which $f$ has the matrix $B$

Suppose and $B_1=(v_1,v_2,v_3)$ and $C_1=(w_1,w_2,w_3)$ are ordered basis of $\mathbb{R}^3$ in which $f$ has matrix representation A=\begin{bmatrix} 1&0&2\\ 7&2&9\\ 1&0&5 ...
1 vote
Accepted

### Inductive proof that any nilpotent operator is upper-triangularisable

The idea is nice, but there is a big hidden induction problem. I translate what you tried to mean in more familiar words to me. Let me know if this was the same your idea. If $T=0$, then it is clear. ...
1 vote

### Prove $ST = 0 \implies ST^* = 0$.

Your proof is correct, and this statement is not true in general for non-normal operators. Let $\{e_0, e_1, \dots\}$ be a basis for your Hilbert space and consider the shift operator $T(e_i) = e_{i+1}$...

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