Elementary matricies (Ex)
Let be an associative ring with unit and recall that an elementary matrix over is a square matrix of the form
where is the identity matrix, and is the matrix with zeros in all places except where it is and we have . Clearly each elementary matrix is invertible and so defines an element where
is the limit of the invertible matricies.
Exercise 0.1. Prove that , where is the subgroup generated by all elements in which are represented by elementary matrices.
Hint 0.2. For and write the matrix
as a product of elementary matrices
where is an matrix.
Recall that is defined to be the abelian group
Exercise 0.3. Prove that .
References
$ and we have $i \neq j$. Clearly each elementary matrix is invertible and so defines an element $E \in GL(R)$ where $$GL(R) = GL_{\infty}(R) : = \text{lim}_{n \to \infty}GL_n(R)$$ is the limit of the invertible matricies. {{beginthm|Exercise}} Prove that $E(R)=[GL(R),GL(R)]$, where $E(R)\subset GL(R)$ is the subgroup generated by all elements in $GL(R)$ which are represented by elementary matrices. {{endthm}} {{beginrem|Hint}} For $A\in GL(m,R)$ and $B\in GL(n,R)$ write the matrix $$ \left( \begin{array}{cc} ABA^{-1}B^{-1} & 0\ 0 & I \end{array} \right) $$ as a product of elementary matrices $$ \left( \begin{array}{cc} I & X\ 0 & I \end{array} \right) = \prod_{i=1}^{m}\prod_{j=1}^{n} (I+x_{ij}E_{i,j+m}) $$ where $X=(x_{ij})$ is an $m\times n$ matrix. {{endrem}} Recall that $K_1(R)$ is defined to be the abelian group $$ K_1(R) : = GL(R)_{ab} = GL(R)/[GL(R), GL(R)].$$ {{beginthm|Exercise}} Prove that $K_1(\Zz) = \{ \pm 1 \}$. {{endthm}} == References == [[Category:Exercises]]R be an associative ring with unit and recall that an elementary matrix over is a square matrix of the formwhere is the identity matrix, and is the matrix with zeros in all places except where it is and we have . Clearly each elementary matrix is invertible and so defines an element where
is the limit of the invertible matricies.
Exercise 0.1. Prove that , where is the subgroup generated by all elements in which are represented by elementary matrices.
Hint 0.2. For and write the matrix
as a product of elementary matrices
where is an matrix.
Recall that is defined to be the abelian group
Exercise 0.3. Prove that .