MAT3040 Advanced Linear Algebra ·

Statement

Let V=M2(R)V=M_2(\mathbb R), the space of real 2×22\times2 matrices. Let WW be the subspace of all upper-triangular matrices, of the form

(ab0d) \begin{pmatrix}a&b\\0&d\end{pmatrix}

with a,b,d∈Ra,b,d\in\mathbb R. Write the coset represented by AA as A+WA+W.

The formula T:V/W⟶V/W,T(A+W)=AT+W \begin{gathered} T:V/W\longrightarrow V/W,\\ T(A+W)=A^{\mathsf T}+W \end{gathered} defines a linear isomorphism, where ATA^{\mathsf T} denotes the transpose of AA.

Proposed proof

For matrices A,BA,B and scalars α,β\alpha,\beta, T(α(A+W)+β(B+W))=T((αA+βB)+W)=(αA+βB)T+W=(αAT+βBT)+W=αT(A+W)+βT(B+W). \begin{aligned} &T(\alpha(A+W)+\beta(B+W))\\ &\quad=T((\alpha A+\beta B)+W)\\ &\quad=(\alpha A+\beta B)^{\mathsf T}+W\\ &\quad=(\alpha A^{\mathsf T}+\beta B^{\mathsf T})+W\\ &\quad=\alpha T(A+W)+\beta T(B+W). \end{aligned} Thus TT is linear.

Moreover, transposing twice returns the original matrix: T(T(A+W))=T(AT+W)=(AT)T+W=A+W. \begin{aligned} T(T(A+W))&=T(A^{\mathsf T}+W)\\ &=(A^{\mathsf T})^{\mathsf T}+W\\ &=A+W. \end{aligned} Therefore TT has an inverse, namely itself. Hence TT is a linear isomorphism. □\square