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Linear Algebra: finding the pivot column in an abstract matrix

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>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.

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