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Dokuz Eylül University Faculty of Business Department of Economics Fall 2007 ECN 221 LINEAR ALGEBRA PROBLEM SET 3 1. Show that if ku= 0, then k=0 or u=0 2. Prove that (-k)u=k(-u)=-ku 3. Show that V=R2 is not a vector space over R with respect to the operations of vector addition and scalar multiplication: (a,b)+(c,d)=(a+c,b+d) and k(a,b)=(ka, kb). Show that one of the axioms of a vector space does not hold. 4. Show that for any scalar k and any vectors u and v, k(u-v) = ku - kv 5. Show that W is not a subspace of V where W consists of all matrices with zero determinant. 6. Show that W is not a subspace of V where W consists of all matrices A for which A2=A. 7. Prove that u + (-u)=0 8. Show that 0u= 0 for any vector any u. 9. For which value of k will the vector u =(1, -2, k) in R3 be a linear combination of the vectors v= (3, 0, -2) and w = (2, -1, -5)? 10. Write the matrix E= 3 1 1 1 0 0 1 1 as a linear combination of matrices A= 1 0 , B= 1 1 and 0 2 0 1 11. As u= (1, -3, 2) and v= (2, -1, 1) are, write w = (1,7, -4) as a linear combination of u and v. 12. Write w = (1,9) as a linear combination of u = (1,2) and v = (3, -1). 13. Write v = (2, -3, 4)) as a linear combination of u1 = (1, 1, 1), u2 = (1, 1, 0) and u3 = (1, 0, 0). 14. Show that the vectors u=(1,2, 3), v=(0,1,2)and w=(0,0,1) span R 3 15. Show that the vectors u=(1,2, 5), v=(1,3,7)and w=(1,1,-1) do not span R3 16. Write the polynmial v=t2+4t-3 over R as a linear combination of the polynomials e 1=t2-2t+5, e2=2t2-3t and e3=t+3. 17. Determine whether or not the vectors (1,-2,1), (2,1,-1), (7,-4,1) are linearly dependent. 18. Show that if two of the vectors v1......... vm are equal, say v1 = v2,, then the vectors are linearly dependent. 19. Determine whether A and B are dependent where 20. Determine whether (1,1,1), (1,2,3) and (2,-1,1) form a basis of R3. 21. Find a basis and dimension of the subspace W of R4 spanned by (1,4, -1, 3) (2, 1, -3, -1) and (0, 2, 1, -5). 1 1 2 0 1 22. Find the rank of A= 2 6 3 3 3 10 6 5 2