1. Given A E Rmx, prove that A A E Rox" is invertible if and only if the columns of A are linearly independent. 2. For A E Rmxn and w1, . .., Wn > 0, characterize the minimizers x* to m 2 minimize E Wi Aijlj - yi TER i=1 j=1 3. Apply by hand the Gram-Schmidt orthonormalization process to the vectors U2 = U3 9 09 4. Prove that any matrix A E Rmxa can be factored as A = LQ where L E Rmxa is lower triangular and Q E Rnx" is orthogonal. 5. Implement in Python an unsophisticated function producing the QR factorization of a matrix based on the Gram-Schmidt process