Let M₃(ℝ) be all real 3×3 matrices and T₃(ℝ) its upper-triangular subspace, with usual operations. To which listed real vector space is M₃(ℝ)/T₃(ℝ) isomorphic?
Correct answer: (A) Real 3×3 skew-symmetric matrices
Explanation
The quotient has dimension 9−6 = 3. Skew-symmetric 3×3 matrices have three independent entries; symmetric and lower-triangular spaces each have dimension six, and trace-zero matrices dimension eight. Finite-dimensional real spaces of the same dimension are isomorphic. **A**.
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