Linear Combinations of Matrices are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Please see the attached file for the fully formatted problems.
Let P3 = ( it is set of all polynomials with coefficients in Z that are at most of degree 3.)
Let A = and B = where , that is = .
(a) Verify that A and B are bases of the Z-module P3.
(b) Compute the change of basis matrices PAB (from the

Can you please explain to me why the columns of the nxn matrix A span R^n when A is invertible? I feel that if matrix A has columns that span R^n, then the inverse of A should likewise share that same characteristic, the spanning. But I'm not sure if that is a sufficient relationship. Can you give an example of matrix A spanning

Gretchen Schmidt plans to buy shares of two stocks. One costs $32 per share and pays dividends of $1.20 per share. The other costs $23 per share and pays dividends of $1.40 per share. She has $10,100 to spend and wants to earn dividends of $540. How many share of each stock should she buy?
Use the form of " AX=B " equation

Q: Find the determinant of these 2x2 matrices.
(a) mat[4 3; -2 9] (b) mat[5 -7; 4 12]
Please see the Word document for a cleaner version of the problem.

2. Use Theorem 5.2.1 to determine which of the following are subspaces of M22.
Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.
(a) If u and v are vectors in W, then u + v is in W.
(b) If k is any scalar and u is any vector in W,

I need some help with these questions:
2. Calculate the linear state space matrices A,B,C and D for system that is described by the state equations, for deviations from uop = [-1, 1]^T, xop = [1,1,0]^T and yop= [0] (see attached file for better formula representation).
3. A linear system is described by its transfer funct