Find the closest point from b 1 1 2 2T to the subspace span

Find the closest point from b = (1, 1, 2, -2)^T to the subspace spanned by (1, 2, -1, 0)^T, (0, 1, -2, -1)T (1, 0, 3, 2)^T. Find the least squares solution to the system Ax = b where A = (1 2 -1 0-2 3 1 5 -1 -3 1 1) and b = (0 5 6 8). The median price (in thousands of dollars) of existing homes in the Minneapolis metropolitan area from 1989 to 1999 was: First find an equation of the least squares line for these data. Then use the result to estimate the median price of a house in the year 2005, and the year 2010, assuming the trend continues. Compute the eigenvalues and eigenvectors of (1 -2/3 1/2 -10/6). Compute the eigenvalues and eigenvectors of (2 -1 -1 -2 1 1 1 0 1) Suppose lambda is an eigenvalue of A. Show that c lambda + d is an eigenvalue of B = c A + dl. When is lambda = 0 is an eigenvalue of a matrix A? Does the matrix (1 -2 0 0 -1 0 4 -4 -1) admit an eigenvector basis of R^3? If not, what is the dimension of the subspace of R^3 spanned by the eigenvectors? Diagonalize the matrix (2 5 5 0 2 0 0 -5 -3) Let A and B be n Times n matrices. Prove that the matrix products AB and BA have the same eigenvalues. Determine whether the following symmetric matrix is positive definite by computing eigenvalues: (4 -1 -2 -1 4 -1 -2 -1 4).

Solution

6. Suppose be an eigenvalue of AB & X ( 0) be the eigenvector of AB corresponding to
By definition of eigenvalue & eigenvector
AB(X) = X
Let Y = BX
As X 0 & B is a non-singular matrix, Y = BX 0
Now,
BA(Y) = BA(BX) = B(AB(X)) = B(X) = B(X) = Y
Hence, is an eigenvalue of BA