>I am confused how we can assume this though, can't column 2 be regarded as a linear combination of column 1 and 3?
It can. However, this becomes irrelevant when answering the question, since columns 1 and 2 are to the left of column 3.
To see this, let A be the matrix with a_1, a_2 and a_3 as columns, and let A^r be the reduced row echelon form of A (we don't need to calculate A^r explicitly). Clearly the pivot columns of A and A^r are the same, and the pivot columns in A^r are the ones that contain the first non-zero element for each row.
Note that since
A(3,2,1)^⊤ = 0
it follows that
A^r(3,2,1)^⊤ = 0.
If we let a^r _1, a^r _2, amd a^r _3 be a row of A^r , we then have
3a^r_1 + 2a^r_2 + a^r_3 = 0
Rearranging we get
a^r_3 = -(3a^r_1 + 2a^r_2),
and thus a^r_3 can't be the first non-zero element in that row.