1. Let A=(a_(i,j)) be a n x n matrix with real coefficients (a_(i,j)) in [0,1[ such that
for all i, sum(a_(i,j),j=1..n) < 1. Show that |det(A)| <= 1.
Solution

2. Let E be an infinite-dimensional complex vector space. Show that you can
construct two norms on E which are not equivalent.
Solution

3. Let K be a compact subset of a normed vector space E. Show that there is a closed separable
subspace H such that K C H.
Solution

4. Given 3 real matrices A,B,C in M_n(R), show that:
$tr(A(A^T-B^T)+B(B^T-C^T)+C(C^T-A^T)) >= 0.
Solution

5. Let E be a normed vector space. Show that it is impossible to find two continuous linear mappings
u,v: E -> E such that uv-vu=id_E.
Solution

6. In a commutative ring R, show that a matrix is invertible if and only if its determinant is a
unit.