Question: An n n matrix A is upper-triangular if: A = a11 a12 a13 a1n 0 a22 a23 a2n 0 0 a33 a3n ... ... ...

An n n matrix A is upper-triangular if: A = a11 a12 a13 a1n 0 a22 a23 a2n 0 0 a33 a3n ... ... ... . . . ... 0 0 0 ann . Let Tn(F) denote the subset of all upper-triangular matrices in Mn(F). (a) Prove that Tn(F) is a subspace of Mn(F). (b) Let Nn(F) denote the subset of Tn(F) of strictly upper-triangular matrices; that is, A Nn(F) if and only if aii = 0 for i = 1, . . . , n. Prove that Nn(F) is a subspace of Tn(F). (c) Let Dn(F) denote the subset of Tn(F) of diagonal matrices; that is, A Dn(F) if and only if aij = 0 whenever i < j. Prove that Dn(F) is a subspace of Tn(F). (d) Prove that Tn(F) = Dn(F) Nn(F). (internal direct sum) Proof.

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