Linear Algebra Show workexplanation Suppose V R4 and W is t

Linear Algebra

Show work/explanation.

Suppose V = R^4 and W is the orthogonal complement to (1 1 1 1) An simple basis for W is given by the three vectors a_1 = (1 -1 0 0), a_2 = (0 1 -1 0), a_3 = (0 0 1 -1) However, this basis is not orthonormal. We get an orthonormal basis q_1, q_2, q_3 as follows. Define C_1 = a_1. Define (and compute) c_2 = a_2 - c_1^T a_2/c_1^T c_1 c_1. Check that c_2 middot c_1 = 0. This is less scary than it seems! We are taking a_2 and removing from it just the right amount of C_1 (i.e. removing some multiple of c_1) to make it orthogonal to C_1. Define (and compute) c_3 = a_3 - c_1^T a_3/c_1^T c_1 c_1 - c_2^T a_3/c_2^T c_2 c_2. Check that c_3 middot c_1 = 0 and c_3 middot c_2 = 0. We know have W = (c_1, c_2, c_3), where the c\'s are orthogonal vectors. As a final step, compute q_1 = c_1/|c_1|, q_2 = c_2/|c_2|, q_3 = c_3/|c_3| to get three orthonormal vectors that span W.

Solution