Problem 1E: Which of the following sets of vectors form an orthonormal basis for 2? {(1,0)T,(0,1)T} {( 3 5, 4... Problem 2E: Let u1=( 1 3 2 1 3 2 4 3 2 ),u2=( 2 3 2 3 1 3 ),u3=( 1 2 1 2 0) Show that {u1,u2,u3} is an... Problem 3E: Let S be the subspace of 3 spanned by the vectors u2 and u3 of Exercise 2. Let x=(1,2,2)T. Find the... Problem 4E: Let be a fixed real number and let x1=( cos sin) and x2=( sin cos) Show that {x1,x2} is an... Problem 5E: Let u1 and u2 form an orthonormal basis for 2 and letubeaunitvector in 2. If uTu1=12,... Problem 6E: Let {u1,u2,u3} be an orthonormal basis for an inner product space V and let u=u1+2u2+2u3 and... Problem 7E: Let {u1,u2,u3} beanorthonormalbasisforaninner product space V. If x=c1u1+c2u2+c3u3 is a vector with... Problem 8E: The functions cosx and sinx form an orthonormal set in C[,]. If f(x)=3cosx+2sinx and g(x)=cosxsinx... Problem 9E: The set S={12,cosx,cos2x,cos3x,cos4x} isanorthonormalsetofvectorsin C[,] inner product defined by... Problem 10E Problem 11E Problem 12E: If Q is an nn orthogonal matrix and x and y are nonzero vectors in n, then how does the angle... Problem 13E Problem 14E Problem 15E: Let Q be an orthogonal matrix and let d=det(Q). Show that |d|=1. Problem 16E: Show that the product of two orthogonal matrices is also an orthogonal matrix. Is the product of two... Problem 17E Problem 18E Problem 19E Problem 20E Problem 21E: Let A=( 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ) Show that the column vectors of A form anorthonormal set... Problem 22E Problem 23E Problem 24E: Let A be an mn matrix, let P be the projection matrix that projects vectors in m onto R(A), and let... Problem 25E: Let P be the projection matrix corresponding to a subspaceSof m. Showthat P2=P PT=P Problem 26E Problem 27E: Let v be a vector in an inner product space V andlet p be the projection of v onto an n-dimensional... Problem 28E: Let v be a vector in an inner product space V and let p be the projection of v onto an n-dimensional... Problem 29E: Given the vector space C[1,1] with inner product f,g=11f(x)g(x)dx and norm f=(f,f)1/2... Problem 30E: Consider the inner product space C[0,1] with inner product defined by f,g=01f(x)g(x)dx Let S be the... Problem 31E Problem 32E: Find the best least squares approximation to f(x)=|x| on [,] byatrigonometricpolynomial of degree... Problem 33E: Let {x1,x2,...,xk,xk+1,...,xn} be an orthonormal basis for an inner product space V. Let S1 be the... Problem 34E Problem 35E Problem 36E: A(real or complex)scalar u is said to bean nth root of unity if un=1. Show that if u is an nth root... Problem 37E Problem 38E format_list_bulleted