This may be a dumb question to ask but, how would one solve a infinite system of linear equations with an indefinite number of unknowns.

I was giving this problem in matrix from:

X1+X3+X5=0
X2+X4+X6=0
X3+X5+X7=0
. . .
. . .

and have to put it in this form

Now after just by looking at it, "X" can be any number. I uses row reduction, even matrix algebra. Though is there a theorem or a formula that would break this down. A pattern in away to solve this. I was looking up matrices online and I was looking for something to help me out. I know how to solve a matrix...just I don't know what being asked to do.

 

Is this a problem in a book on linear algebra? If so, is there any other information that's given or hints or anything? An obvious solution is x1 = x2 = x3 = ... = xn = ... = 0.

If the matrix of coefficients is invertible, then the solution above is the only solution. Since the matrix of coefficients is upper triangular (i.e., all of the coefficients below the main diagonal are zero), then I think it's possible to show that the matrix of coefficients has an inverse, and therefore, the solution vector is (0, 0, 0, ..., 0, ...).

Does that help?