For nge3, consider the space Mn(C) of ntimes n complex matrices endowed with the inner product langle A,Brangle=Trace(A^B). For 0le kle n, let Wk=Ain Mn(C):langle A,Brangle=0 for all Bin Mn(C) with rank k. Which of the following statements are necessarily true?

Q78Section C (Part C)78 of 120

Matrix spaces — which ranks span under the trace inner product

For $n\ge3$ , consider the space $M_n(\mathbb{C})$ of $n\times n$ complex matrices endowed with the inner product $\langle A,B\rangle=\operatorname{Trace}(A^{*}B)$ . For $0\le k\le n$ , let

$W_k=\left\{A\in M_n(\mathbb{C}):\langle A,B\rangle=0 \text{for all }B\in M_n(\mathbb{C})\text{ with rank }k\right\}.$

Which of the following statements are necessarily true?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Select every correct option, then submit.

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