Answer Happy

1. (a) [4 points) Find the solution space S of the system of linear equations Ax= 0, where 21 22 A= 1 3 -1 2 --3 5 -4 -11 -5 -12 23 34 25 (b) [11 points) Let S be the solution space that you found in Part (a). Use the Gram-Schmidt process to find an orthonormal basis B for S. Note: Let S be a subspace of some vector space. A set of vectors B = {bi,..., bp) is an orthonormal basis for S if B is an orthogonal basis for S and the vectors in B are unit vectors. (c) [3 points) Let B {b},..., bp} be an orthonormal basis for some subspace S. Let x € S. Show that x= cibi +...+ Cp bp, where Cj = x•bję for j = 1,..., p. (d) [2 points) Let S be the solution space that was found in Part (a) and let B be the orthonormal basis for S that was found in Part (b). Define -2 1 2 -1 1 ES. Write x as a linear combination of the vectors in B. Hint: Use the result in Part (c).