b) Suppose that the augmented matrix for a linear system has been reduced by row operations to the given row echelon form. Determine if a solution exists. If a solution does exist, find all solutions 1 -3 7 1 0 1 4 0 0 00 1 Justify your answer. a) Write the following systems of equations as a single matrix equation of the form Ax b. Give A, x, and b. 4x1 + 2x3 = 1 3x1-22=3 b) Determine whether or not the set {(1, 1,0), (0, 1, 1), (1, 0, -1)} is a basis for R³. Consider the following basis B = {V1, V2, V3} of R³: 1 1 V1 √3 " V2 √2 (§) V3= -(e) For x, y € R3, the inner product is defined as (x, y) =xy = 131+222 + 3ys, a) Show that (V1, V1) = (V2, V2) b) Show that v₁ is orthogonal to V2. -9)-B = y= Y2 Уз c) Use the Gram-Schmidt process to find v3 so that the set B = {V1, V2, V3} forms an orthogonal basis of R3. Write vз in terms of v3, (V1, V3)V₁ and (V2, V3)V2) Consider the basis B = {1+x,1-x} and E = {1,2} of P₁, the vector space of all polynomials of degree 1 or less. Let the linear transformation T: P₁→ P₁ be defined as a) Find the matrix PB-E. T(p(x)) = xp'(x). b) Find the matrix [T]B,B = [T] B. c) Find the matrix [T]E,E = (TE