This is the 14th problem in the 3rd section of the book: "Introduction to Linear Algebra" by Gilbert Strang.
The problem states: "Suppose Column 1 + Column 3 + Column 5 = 0 in a 4 by 5 matrix with 4 pivots. Which column has no pivot? What is the special solution? Describe N(A)."
The solution is supposed to be: "Column 5 is sure to have no pivot since it is a combination of earlier columns. With 4 pivots in the other columns, the special solution is s = (1, 0, 1, 0, 1). The nullspace contains all multiples of this vector s (this nullspace is a line in $R^5$).
I have no idea why column 5 would be free column, since while values in matrix of columns 1, 3, and 5 depend on each other I don't see a reason variables should(with variables I mean the matrix we are multiplying with, when finding N(A). Is this not the correct interpretation, and what am I missing?
Thank you for your help!