Let $$S:=\{AB-BA:A,B\in M_3(\mathbb R)\}\leq M_3(\mathbb R)$$ Find $\dim S$.
Can anybody help me?
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1Notice that the matrices $AB - BA$ all have zero trace. – Michael Biro Jan 31 '20 at 15:14
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Does this answer your question?https://math.stackexchange.com/questions/2677496/subspace-of-the-real-matrices-generated-by-the-matrices-of-the-form-ab-ba – Matcha Latte Feb 01 '20 at 12:23
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Also $\underline{\text{traceless matrices}}$ :https://math.stackexchange.com/q/729796/721644$$$$ https://projecteuclid.org/euclid.mmj/1028990168 – Matcha Latte Feb 01 '20 at 12:26
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Look at the number of free parameters under the constraint $\mathrm{tr}(s) = 0$ for $s \in S$ – Mathphys meister Feb 01 '20 at 14:57