Need help with problem 1, 2, 3, and 4.

1. Verify that a matrix multiplied by a column vector is simply a linear combination of the columns of the matrix. Namely, if the columns of the matrix A are a_1, a_2, .. a_n and the vector x is (x_1,... x_n), then Ax = x_1 a_1 + ... + x_n a_n. Explain why this is simply the outer product way of multiplying A and x.

2. Now verify that a row vector multiplied by a matrix is simply a linear combination of the rows of the matrix. Explain again that this is just the outer product way of multiplying.

3. Using the insights in 8: obtain a matrix P such that if A is any matrix with 3 columns, AP is a cyclic shift of the columns of A (namely the first column of A is the second column of AP, second column of A is the third column of AP, and the third column of A becomes the first column of AP).

4. Using the insight in 9: obtain P such that if A is any matrix with 3 rows, PA is a cyclic shift of the rows (cyclic as explained in 10.)