Discussion
Let \(M_3(\mathbb{R})\) be the vector space of all \(3 \times 3\) real matrices over \(\mathbb{R}\) under usual matrix addition and scalar multiplication. Let \(T_3(\mathbb{R})\) be the set of all \(3 \times 3\) real upper triangular matrices. Which one of the following is TRUE?
Answer: A
The quotient is isomorphic to the space of 3 by 3 real skew symmetric matrices.
Finite-dimensional real vector spaces are isomorphic exactly when their dimensions are equal. \(M_3(\mathbb{R})\) has dimension 9 and the upper triangular matrices have dimension 6 (the diagonal and the three entries above it), so
\[ \dim M_3(\mathbb{R})/T_3(\mathbb{R}) = 9 - 6 = 3. \]
Skew symmetric 3 by 3 matrices are fixed by the three entries above a zero diagonal, so they also have dimension 3. Symmetric and lower triangular matrices each have dimension 6, and trace-zero matrices have dimension 8, so none of those is isomorphic to the quotient.
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