Sample Quiz on Inner Product Spaces Question # 1: If B= {v1,v2,v3} is an orthogonal set of vectors with respect to an inner product on a vector space V, then the set B Select Answer Here (a) is linearly independent (b) is linearly dependent (c) is an orthonormal basis for V (d) spans the vector space V Question # 2: Which one of the following unit vectors u1:=vector([0,1/sqrt(2),1/sqrt(2)]);u2:=vector([0,-1/sqrt(2),1/sqrt(2)]) u3:=vector([0,1/sqrt(3),-1/sqrt(3)]);u4:=vector([0,-1/sqrt(3),1/sqrt(3)]) is orthogonal to the vectors v1:=vector([1,-1,1]) and v2:=vector([1,0,0])? Select Answer Here (a) vector u1 (b) vector u2 (c) vector u3 (d) vector u4 Question # 3: The orthogonal projection of the vector v:=vector([1,-1,1]) upon u:=vector([1,1,1]) Select Answer Here (a) is the vector u1:=vector([1/sqrt(3),1/sqrt(3),1/sqrt(3)]) (b) is the vector u2:=vector([1/sqrt(3),-1/sqrt(3),1/sqrt(3)]) (c) is the vector u3:=vector([-1/3,1/3,1/3]) (d) is the vector u4:=vector([1/3,1/3,1/3]) Question # 4: If w is the orthogonal projection of the vector v upon the vector u where v:=vector([1,-1,1]); u:=vector([1,1,1]); w:=vector([1/3,1/3,1/3]). Then the orthogonal complement Select Answer Here (a) is the vector u1:=vector([1/3,4/3,1/3]) (b) is the vector u2:=vector([2/3,-4/3,2/3]) (c) is the vector u3:=vector([2/3,-2/3,2/3]) (d) is the vector u4:=vector([2/3,2/3,2/3]) Question # 5: The cosine of the angle between the two vectors v and u v=vector([1,-1,1]); u=vector([1,1,1]) Select Answer Here (a) is equal to 1/3 (b) is equal to 1/9 (c) is equal to 1/sqrt(3) (d) is equal to -1/3 Question # 6: The inverse of the matrix A: A:=matrix([[1/sqrt(2),-1/sqrt(2)],[1/sqrt(2),1/sqrt(2)]]); A1:=matrix([[1/sqrt(2),-1/sqrt(2)],[1/sqrt(2),1/sqrt(2)]]); A2:=matrix([[1/sqrt(2),1/sqrt(2)],[-1/sqrt(2),1/sqrt(2)]]); A3:=matrix([[1/sqrt(2),-1/sqrt(2)],[-1/sqrt(2),1/sqrt(2)]]); A4:=matrix([[-1/sqrt(2),1/sqrt(2)],[1/sqrt(2),-1/sqrt(2)]]); Select Answer Here (a) A3 (b) A1 (c) A2 (d) A4 Question # 7: If the vectors v1, v2 and v3 form an orthonormal basis for R3 v1:=vector([1,0,0]); v2:=vector([0,1/sqrt(2),1/sqrt(2)]); v3:=vector([0,-1/sqrt(2),1/sqrt(2)]); Then The vector w:=vector([1,-1,1]) can be expressed as a linear combination of the vectors v1, v2 and v3 in the following way: Select Answer Here (a) w = 1*v1 +2/sqrt(2)*v2 + 2/sqrt(2)*v3 (b) w = 1*v1 + 2/sqrt(2)*v2 (c) w = 1*v1 + 2/sqrt(2)*v3 (d) w = 1*v1 - 2/sqrt(2)*v2 +2/sqrt(2)*v3 Question # 8: The vectors v1, v2 and v3 form an orthonormal basis for R3 where v1:=vector([0,0,-1]); v2:=vector([1/sqrt(2),1/sqrt(2),0]); v3:=vector([-1/sqrt(2),1/sqrt(2),0]); The coordinates of the vector w:=vector([1,-1,1]) with respect to the basis vectors v1, v2 and v3 of R3 are: Select Answer Here (a) -1, 0, -2/sqrt(2) (b) 1, -2/sqrt(2),0 (c) 1, 2/sqrt(2),0 (d) -1,- 2/sqrt(2), 2/sqrt(2)