Page < 2 4) Determine whether each of the following sets, together with the standard operations, is a vector space. If it is, then simply write 'Vector space'. You do not have to prove all ten vector space axioms. If it is not, then identify one of the ten vector space axioms with its number in the attached sheet that fails and also show that how it fails. a) The set of all polynomials of degree four or less. b) The set of all 2 x 2 singular matrices. c) The set {(x, y) : x≥ 0, y is a real number} d) C[0,1], the set of all continuous functions defined on the interval [0,1] of 2 ZOOM + Page < 1 > of 2 ZOOM + 1) a) Answer the following questions by circling TRUE or FALSE (No explanation or work required). [1 0 0 i) A = 0 2 6 is invertible. (TRUE FALSE) LO -4-12] ii) We can use the transpose of the cofactor matrix to find the inverse of a matrix. (TRUE FALSE) = iii) If A 2, and A is a 5x5 square matrix, |2A] = 64. (TRUE FALSE) iv) Every vector space must contain two trivial subspaces. (TRUE FALSE) v) The set of all integers with standard operations is a vector space. (TRUE FALSE) b) Write v as a linear combination of the vectors in the set S, if possible, where v=(1,-4), and S={(1,2),(1,-1)}. 2) a) Solve the following system of linear equations using Cramer's Rule and check the correctness of your answer. 4xyz 1 2x + 2y + 3z = 10 5x-2y-2z = -1 b) Find the adjoint of the following matrix A. Then use the adjoint to find the inverse of A if possible, and check the correctness of your answer. A = c) Determine whether the following points are collinear. Why or why not? If not, then find the area of the triangle whose vertices are these points. (1,1), (2,4), and (4,2) 3) a) Describe the zero vector and the additive inverse of a vector in the vector space, M2,3. b) Determine if the following set S is a subspace of R² with the standard operations. Show all appropriate supporting work. S = {(x, y) : x ≥ 0, y ≥ 0} c) Determine if the following set S is a subspace of M3,3 with the standard operations. Show all appropriate supporting work. The set of all 3 x 3 upper triangular matrices 4) Determine whether each of the following sets, together with the standard operations is a vector space. If it is then simply write Vector space' You do not