Science talk:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (c)

I would like to see a short explanation of why

This time, row operations (or left multiplication) changes the column space, it still preserves the relations among the columns.

Also, This time is unfortunate wording for a timeless fact. If you want to contrast this fact to (a) and (b), please be more specific (maybe tie it to the reason for the main claim).

Thank you for making the reasoning very clear. I added another sentence to round up the argument. Please make sure that it's consistent and then flag accordingly.

Since my answers for the other questions were found to be lacking, I am verifying them using sage. So for this question:

A = Matrix(QQ, 4,5,[[1,2,1,3,2],[3,4,9,0,7],[2,3,5,1,8],[2,2,8,-3,5]])

v1 = vector(QQ,[1,3,2,2])

v2 = vector(QQ,[2,4,3,2])

v3 = vector(QQ,[3,0,1,-3])

span([v1,v2,v3],QQ) == A.column_space() // This evaluates as true, so done.