Please answer questions 3, 5, 6, 7, and 8 only.

U1: 1 ,022 1 #232 O 5 1 3 Find the transition matrix from B to B'. What could be the transition matrix from B' to B? 11. Are these three vectors in R4 U1 2 (1,3,5,1),u2 = (0,2,1,3),u3 = (2,3,0, 1) linearly independent? 12. Show that rank(A) = rank(AT). 13. Show that if rank(A) = 0, then A = 0, and that if rank(Aan) = n, then A is invertible. The reverse is also true. 14. Given 1 3 4 2 5 4 l 2 6 9 1 8 2 l 2 6 9 1 9 7 [1 3 4 2 5 4J and its row echelon form: 1 3 4 2 5 4 0 0 1 3 2 6 0 0 0 0 1 5 0 0 0 0 0 0 a. Find the basis of R0w(A) b. Find the basis of Col(A) c. the rank of A 15. Given A in Problem 14, nd the null space of A with basis and its nullity. Conrm the identity between the rank and nullity. 16. Given three vectors : u,v,w in a vector space, show span{u,v,w} = span{u,u + 1), u + w} 17. Show W = {A : AT = 5A} forms a subspace in MM". What is the dimension of W? 18. If m, .., on are independent, then so are 02, v3, mun. 19. If m, .., on are dependent then so are 01, ..., Umw for any other vector in. Test2-review Chapter 3 and 4 Prof. Pan's Math 351 Name. 1. Given u = (2, 1,3,4) in R4 nd two vectors of length of 10 in the direction of u and the opposite direction of u. 2. Given two vectors u = (1,2,4,1),v = (2,3,0,1) in R4 , nd the angle between them. 3. Given u = (2, 1, 3,4) in R4 , nd all vectors that are orthogonal to u. Show these vectors form a subspace of R4 , what is the dimension ? Find a basis as well. 4. Given two vectors u = (2, 2, 0, 1),v = (2, 3,0, 3) in R4 , nd Pmqu and Projuv respectively, and sketech them. 5. Determine which of the following are subspaces of M "X\" The set of all n X n matrices such that det(A) = 0. . The set of all n x n matrices such that AT 2 A: symmetric matrices. The set of all n X n matrices such that AT 2 A: anti-symmetric matrices. Determine which of the following are subspaces of P3, polynomials of degree at most @9693 All polynomials a0 alzz: a2m2 : a3x3 for which a0 = 0. . All polynomials a0 + alx + tax? + a3x3 for which a0 + a1 + a2 + a3 = 0. . All polynomials a0 alzz: (12:172 + (13.173 for which a2 = 2 . Show that the following polynomials form a basis for P3: 1 + :13, 1 :12, 1 x2, 1 :1:3 . Show that the following matrices form a basis for M 2\". [:5 1113), 32113131133] 9. Consider the basis B = {u1,U2,U3} for R3 where ooqoo'y 1 0 1 U1: 1 ,U22 1 ,U32 0 1 1 1 Find the coordinate map for this basis, and compute the coordinates of the vector 2 U=2 2 10. Consider the bases B = {111,112,113} and B' = {121,122,123} for R3 where 2 2 1 U1: 1 ,U22 1 ,U32 2 1 1 1