5. The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space. The method works as follows: if {(11} a2, . . . , {IN} is a set of linearly independent vectors in RM (so clearly N g M) then we can generate a sequence of orthonomal vectors {q1, q2, . . . , q N} such that Spanah - - - ,GNH = Spanqh - - -,qN}) using a1 9' Z 1 Halll2 and for k22,...,N: kil wk = We 2011c: [Rhea {3:1 wk (1 = - k IIWk||2 4 Last updated 15:56, October 21, 2022 (a) As a warm-up, nd '11: qg and (13 when 1 1 1 1 1 1 a1 = 1 a2 = 1 , a3 = 1 1 1 1 1 1 1 Feel free to use a computer to do the calculations; just explain what you did in your write-up. (b) For {(11, a2, a3} and {q1, q2, 9'3} as in part (a), let I I l l l l A = a1 a2 a3 : Q = 9'1 (12 (1'3 1 Show how we can write A : QR, where R is upper triangular. Do this by explicitly calculating R. (Hint: just keep track of your work while doing part (51))- (c) Suppose I run the algorithm above on a general M X N matrix A with linearly independent columns (full column rank). Explain how the GramSchmidt algo- rithm can be interpreted as nding a M X N matrix Q with orthonormal columns and an upper triangular matrix R such that A : QR. Do this by explicitly writing what the entries of R are in terms of the quantities that appear in the algorithm. This is called the QR decomposition of A