I need help improving my discussion question. I cant seem to figure out a good response!

Here is what I have written so far:

Hey Casey, Great job on this week's discussion post. I got the same vector space as you did for this response I am going to verify axiom 2

A_mxn, the set of all mxn matrices.

Axiom 2: u+v=v+u

Lets start out by settingu=[a_ij] be a mxn matrix (where 1<= i<= m and 1<= j<= n) and lets also setv=[b_ij] be another mxn matrix. Then we are going to use properties of addition of matrices to showu+v=[a_ij+b_ij]=[b_ij+a_ij] =v+u.

Next we are going to Let A_mxnbe the set of mxn matrices.

An element of A_mxnis a matrix X=[a_ij] where the real number a_ijis the element of X in position (i, j) and where 1<= i<= m and 1<= j<= n.

To show thatu+v=v+uletu=[a_ij] be a mxn matrix (where 1<= i<= m and 1<= j<= n) andv=[b_ij] be another mxn matrix. Then

u+v=[a_ij] +[b_ij] =[a_ij+b_ij]

but for all i, j we have a_ij+b_ij=b_ij+a_ijbecause a_ijand b_ijare real numbers. So

u+v= [a_ij+b_ij] = [b_ij+a_ij] = [b_ij] +[a_ij] =v+u.

The learning module this week introduces the concept of a vector space.A vector space is a collection of vectors that satisfy 10 axioms.In the graded discussion forum this week you'll verify that axioms of the candidate vector spaces listed below do (or do not) hold.Note that some of the candidate vector spaces are indeed valid vector spaces so all 10 of their axioms will hold.Other candidate vector spaces listed below may not be actual vector spaces so some of their axioms will fail.

Response Post:

For your response post this week you only need to respond toONEstudent discussion where you'll analyze an axiom thathasn't yet been verified.You can select the discussion board post to respond to and additional axiom to analyze, just make sure to select an axiom that hasn't yet been analyzed.

Candidate Vector Spaces

1. Rⁿ, the set of all dimension-n vectors forany n >= 1.
2. A_mxn, the set of all m x n matrices.
3. P_n, the set of all polynomials of degree <= n.
4. C[a,b]the set of all continuous functions defined on the closed interval [a,b].
5. A subspace of R²: the set of all dimension-2 vectors [x; y] whose entries x and y are odd integers.
6. A subspace of R²: the set of all dimension -2 vectors [x; y] such that x>=0 and y>= 0.Note that this is just the 1^stquadrant of the 2-D plane.
7. A subspace of R³: the set of all dimension -3 vectors [x; y; z] such that x+y+z = 0 where x, y, and z are real-valued numbers.

Here is his discussion post:

For this week, I was assigned the vector space (2):

A_mxn, the set of all mxn matrices.

and the axiom (7):

c (u + v) = cu + cv

wherecis a scalar andu A_mxnandv A_mxn.

This is a property of all matrices, where c (u + v) = cu + cv; for all scalars c.

In this case, all vectors are elements ofA_mxn,making them equal dimensions and also adhering to the distributive property of matrices where A(B+C) = AB + AC. Except in this case, A is not another matrix, but a scalar distributed to both mxn matrices.