Problems:

1. For vectors a and b, we have ?a?=6, ?b?=3. What are the largest and smallest values possible for ?a-b? and aTb ?
2. Prove that the matrix A and [A AB] have the same column space.
3. Suppose A is a 5*4 matrix with rank 4. Prove that Ax = b has no solution when the 5*5 matrix [A b] is invertible, and Ax = b is solvable when [A b] is singular.
4. Suppose A is a 3*3 matrix and it satisfies AT=-A. Prove that det(A) = 0.
5. (Matlab) Given 13 two-dimensional data points as: x = [0 2 4 6 8 10 12 14 16 18 20 22 24], y = [15 14 14 16 20 23 26 27 26 25 22 18 16]. Do the least square regression fit the data points with n-order polynomial. You have to decide what order (n) you would use and the reason of your decision. Write down the formulation of calculation. Please also submit your Matlab code and fitting result (the parameters, the plot of the polynomial, and the least square error).

Problems: 1. For vectors a and b, we have \"a\" = 6, Mb\" = 3. What are the largest and smallest values possible for \"a E)\" and aTb ? Prove that the matrix A and [A AB] have the same column space. Suppose A is a 5*4 matrix with rank 4. Prove that Ax : b has no solution when the 5*5 matrix [A b] is invertible, and Ax : b is solvable when [A b] is singular. Suppose A is a 3*3 matrix and it satisfies Ar = A. Prove that det(A) : 0. (Matlab) Given l3 two-dimensional data points as: X: [0 2 4 6 81012141618 20 22 24], y=[15 141416 20 23 26 27 26 25 22 l8 16]. Do the least square regression t the data points with norder polynom1al You have to decide what order (n) you would use and the reason of your decision. Write down the formulation of calculation. Please also submit your Matlab code and tting result (the parameters, the plot of the polynomial, and the least square error)