11 September 2014
1) Which of the following sets of vectors α=(a1 ,a2,a3,…….,an) in Rn are subspaces of Rn (n≥3)?
- All α such that a1≤0;
- All α such that a3 is an integer;
- All α such that a2+4a3=0;
- All α such that a2 is rational.
2) Let C be the field of complex numbers and let n be a positive integer (n≥2).Let V be the vector space of all n×n matrices over C. Which of the following sets of matrices A in V are subspaces of V?
- All invertible A;
- All non-invertible A;
- All A such that AB=BA,where B is some fixed matrix in V.
3) Prove that the union of two subspaces is a subspace if and only if one is contained in the other.
