**Title: QR Factorization Using Modified Gram-Schmidt Orthogonalization****Objective:**To find the full QR factorization of a given matrix using the modified Gram-Schmidt orthogonalization process.**Problem Statement:**Apply modified Gram-Schmidt orthogonalization to find the full QR factorization of the matrix:\[ A = \begin{bmatrix} 4 & 8 & 1 \\ 0 & 2 & -2 \\ 3 & 6 & 7 \end{bmatrix} \]**Explanation:**In the modified Gram-Schmidt process, the matrix $ A $ is decomposed into two matrices: $ Q $ and $ R $. Here, $ Q $ is an orthogonal matrix, and $ R $ is an upper triangular matrix. This method iteratively orthogonalizes the columns of $ A $ and ensures numerical stability compared to the classical Gram-Schmidt process.**Procedure:**1. **Initialization:** Start with the first column of $ A $.2. **Orthogonalization:** Iteratively project each successive column against the previously computed orthogonal vectors.3. **Normalization:** Divide each orthogonal vector by its norm to form the columns of $ Q $.4. **Construction of $ R $:** The elements of $ R $ are determined during the orthogonalization process.The process ensures that each column of $ Q $ maintains orthogonality with respect to the others, thus preserving the orthogonal invariants, while the matrix $ R $ captures the coefficients required to reconstruct $ A $ as $ A = QR $.This approach is particularly useful in numerical linear algebra for solving linear systems, eigenvalue problems, and more, where maintaining numerical accuracy and stability is crucial.

Given matrix is A=48102-2367 Let β1=4,0,3 , β2=8,2,6 , β3=1,-2,7 Now assume that S={α1,α2,α3} is orthonormal basis of A then we will evaluate this by using Gram-Schmidt orthogonalization process as follows Step(1): α1=γ1γ1 where γ1=β1=4,0,3 , then γ1=γ1,γ1 =42+02+32 = 25=5 Thus α1=4,0,35 ⇒ α1=45,0,35 Also take r11=γ1=5 Step(2) : α2=γ2γ2 where γ2=β2-β2,α1α1 ⇒ γ2=8,2,6 - 458+02+35645,0,35 ⇒ γ2=8,2,6 - 1045,0,35 ⇒ γ2=0,2,0 γ2=γ2,γ25=02+22+02=2 Thus α2=γ2γ2=0,1,0 Also r22=γ2=2 and r12=β2,α1=10 Step(3) : α3=γ3γ3 where γ3=β3-β3,α2α2-β3,α1α1 γ3=1,-2,7 -10+-21+700,1,0-1514+-20+7345,0,35 γ3=1,-2,7 --20,1,0-125254,0,3 γ3=1,-2,7 -0,2,0-4,0,3 γ3=-3,-4,4 γ3=γ3,γ3=41 Thus α3=γ3γ3=-341,-,-441,441 Also we have r33=γ3,γ3=41 , r13=β3,α1=1 and r23=β3,α2=-2 Finally the orthonormal matrix Q=450-34101-441350441 and R=r11r12r130r22r2300r33=510102-20041 Observe that Q is an orthonormal matrix and R is an upper triangular matrix.