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As you know there is a possibility of an after test party. I will organize one in my house on Thursday from 6 to 9 p.m. if there is enough interest. Music, basketball, pizza and drinks will be provided. If you need more, bring it with you. Please, tell me whether you plan to go to the party. I will post more information if the party will take place.

Here are the problems.

1. Prove that similar matrices have the same traces.
2. Prove that if T is an invertible linear operator in a vector space V and {b_1,...,b_n} is a basis in V then {T(b_1),...,T(b_n)} is a basis in V.
3. Represent the matrix
   4 1 1
   1 4 1
   1 1 4
   as a product of elementary matrices.
4. Find the determinant of the following n by n matrix:
   1 1 .... 1 2
   1 1 .... 2 1
   ............
   1 2 .... 1 1
   2 1 .... 1 1
   
5. Find the rank of the following matrix depending on the parameter t:
   

          t 2t 3t 4t 
          t 2t 3t 4 
          t 2t 3  4
          t 2  3  4 
   

   
   
6. The angle between a vector b and a subspace V in a Euclidean vector space is (by definition) the angle between b and the projection of b on V. Find the angle between the function x^3 and the subspace of C[0,1] spanned by functions 1, x, x^2.
7. Let T be the linear operator in R^2 with the following standard matrix B:
   3 4
   4 3
   Find a basis in R^2 where this operator has a diagonal matrix. Find a matrix M such that M^-1*B*M is a diagonal matrix.
8. Let T be the linear operator in R^3 with the following standard matrix:
   2 1 0
   -1 0 0
   2 2 3
   Does T have a basis of eigenvectors?