Problem 1. (4 X 2 points) For each of the statements below, determine whether it is true or false with justications (eg. if a statement is false, give explicit counterexamples and if it is true, give appropriate arguments). (i) Two distinct matrices A,B can have same eigenvalues and same set of linearly independent eigenvectors corresponding to those eigenvalues. (ii) Suppose that the GramSchmidt process is applied to the set 3 : {ul - - - ,uk} of n-dimensional vectors and it yields an orthogonal set E\" 2 {V1, - ~- ,vk}. If V}, = 0, then uk is a redundant vector in 15'. (iii) Suppose that A is a 3 X 3 symmetric matrix, i.e. AT : A. If Ax : x, Ay : U,Az : 72 for three nonzero vectors x, y, z in R3, then the vector in span{x, 2} that is closest to y is x + 2. (iv) If {V1, - .. ,vn,1,vn} is a basis of R\Problem 3. (8 + 4 points) (i) Find an orthonormal basis {v1, V2, v3} of R3 such that v1, v2 spans the column space of the following matrix A = 2 (ii) What is the least square solution to Ax = b if b = (1 2 7) T