Find a Spanning Sets for a space AX=(A^T)X

[n3kt]

1. The problem statement, all variables and given/known data

Find a spanning set for the space

T(A) = {X in R5 : AX=(A^T)X} , where A^T (A transpose)

A = 58, -20, -4, -35, 34
-20, 58, 31, 1, -36
-4, 31, 43, 7, -21
18, 18, -17, 31, -12
34, -36, -21, -27, 69

2. Relevant equations



3. The attempt at a solution

I don't understand what this question is really asking for...
What I understand is a spanning set is a set that be written as a linear combination. For ex.
{ X1, X2,...Xk} can be written as aX1+bX2+...cXk.

Since I know by a nxn matrix, I took the determinant of A^t) which was not 0, so the column matrix was linearly independent. So i just wrote the spanning set as the Column vectors.

[HallsofIvy]

Not all vectors in R5 satisfy AX= ATX but it is not difficult to prove that the set of vectors that do form a subspace of R5. What they are asking for is a set of vectors that span that subspace. Yes, A and AT here are invertible matrices. But that only means their column vectors (v= Ae where e is a basis vector for R5) span all of R5.

Does the problem really say "find a spanning set" and not a "minimal spanning set" or basis? If so then any set of vectors that span all of R5, such as the columns of either A or AT, or just {<1, 0, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0, 0>, <0, 0, 0, 1, 0>, < 0, 0, 0, 0, 1>} span that subset!

[n3kt]

Yea the question really ask for find a spanning set.

[n3kt]

Thank you, now I understand what the question is asking for...but how can i approach the question now??

[HallsofIvy]

By giving any one of the three answers I just gave you!

[n3kt]

I tried those answers, and type it in in my online assignment, and it says its wrong. So i'm guessing we have to find the subset