i need help with question 39 19. Let A = 0 (i) Find the characteristic polynomia

Question

i need help with question 39

19. Let A = 0 (i) Find the characteristic polynomial of A (i) Find the eigenvalues of A Determine whether S = {(ri,22): 2 1,22 e R and -z-0} is a subspace of the vector space R2 20. 21. Determine whether the vectors (1,-1,3), (2,1,-1), -1,-2,4) are linearly independent in RS 22. Determine whether the set S(1,-2 23. Determine whether S = {(21,22): x1,x2,E3 € R and xit 4x2-5x3 = 0} is a subspace of the 1)) spans the vector space R2 vector space R 24. Determine whether S = {(x1,x2,P2) : x1,x2 R and z2-6#x = 0} is a subspace of the vector space R3 25. Determine whether S = {(z1,22,T2) : x1, z2,23 R and1-4+ 6z2 = 0} is a subspace of the vector space R 26. Determine whether the vectors (1,2,3),(-1,1,-1),-1,4,1) are linearly independent in R 27, Determine whether the vectors (1,-1, 1)·(1,2-1), (-2.-1,-1) are linearly independent in R3 28. Determine whether the vectors (1,2,3), (-11,-1), (-1,4, 1) span R3 29. Determine whether the vectors (1,-1,1), (1,2,-1), (-2,-1,-1) span P3 30. Let B-((1,3,4). (-3,1,-1)). Show that B is a linear independent set. Does B span 3? Complete (add some appropriate vector to) B so that it may be a basis for 31. Determine whether the given vectors form a basis of P 32. Find the coordinates of the vector relative to the basis S = {ul,u2} of R2 ss. Determine whether s-cPa-2n·0)..subpace of R. 33. Determine whether S- 34. Find the coordinates of the vector w relative to the basis s--(11,242} of R2 u.-(1,1), u2-(-1,1) and w-(-4,2). 35. Determine whether the given vectors are linearly independent. P1-1+1+22 , p2 = 1 + x, P3=-x2 + x-3 36. Express the vector p = 1-312 as a linear combination of 37. Prove or disprove that {P1,P2,P3} forms a basis for [x], where [a] is the vector space of real polynomials with indeterminate x, of degrees at most 2 and Pi = 1-2+2, p2-1 + 2x, p3 =-3+z. 38. Determine whether T : R2-R defined by T(x1,x2) = 3x1-2x2 is a linear transformation. 39. Prove or disprove that -ad - c+b is linear tranformation

Explanation / Answer

We let c=b=0 in following example

Let, M be a matrix with a=1,d=0

and N be a matrix with a=0,d=1

T(M)=0

T(N)=0

M+N has, a=1,d=1

T(M+N)=1

Hence, T is not linear

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