1 Consider the matrix A = 7 -3 -5 1 (a) By using Gram-Schmidt process, find an orthogonal basis for Col(A). (b) Find a QR-factorization of A. All computation must be shown (computation of the columns of Q, computation of the matrix R, the check of A = QR).

How is the basis {lk} picked?

(i) Give an example of two orthonormal vectors in R³. (ii) Using C=9 D=3 E=5 F=4 Schmidt process to find an orthonormal basis from the initial vectors () and (7). use the Gram- (iii) Construct a matrix which implements a shear of factor 3 along the first vector of your basis constructed in (ii), in that basis. Then use change of basis methods to give the corresponding (similar) matrix in the standard basis of R². Homework

Find a and that solve the vector equation: Suppose the above equation is expressed as Ax = b with x = coeffieints of the equations in one matrix form as (Ab)= Rows: 2 Columns: 3 b) c) d) 21 a (²) + ^ ( ;) = (²1) B α= B = Complete the missing column of the following matrix derived by doing row operation of the augmented matrix in a): 1 -2 (²). 0 1 Express the

Compute the standard basis of Nul(A) for the matrix (No need to show work for calculating rref)

Qa) attached below Qb) attached below Qc) ? (alpha) = ? Qd) ? (beta) = ?