Using RowReduce in Wolfram-Alpha gives wrong answer

Question

I tried to check whether the vectors [1,1,1],[4,3,2],[3,1,6] are linearly independent over GF(7).

I used "rowreduce([[1,4,3],[1,3,1],[1,2,6]]) in gf(7)" in Wolfram-Alpha and it told me they're independent. However, 3*[4,3,2]+2*[3,1,6]+3*[1,1,1]=0, which implies they're dependent.

I'm trying to figure out what I'm missing.

Would appreciate your help. Thanks.

Answer 1

The proper syntax in Mathematica is:

RowReduce[{{1, 4, 3}, {1, 3, 1}, {1, 2, 6}}, Modulus -> 7]

which yields

{{1, 0, 2}, {0, 1, 2}, {0, 0, 0}}

However, this input is not accepted in WolframAlpha. What it appears to do is perform the row reduction in $\mathbb Z$, then take the result modulo $7$, which is not what you want. Interestingly, if you write

RowReduce[{{1, 4, 3}, {1, 3, 1}, {1, 2, -1}}]

you do see that the matrix is not full rank.

This is an example of how the natural language processing in WolframAlpha does not always give the intended output: whereas Mathematica provides an unambiguous syntax for specifying a particular expression to be evaluated, not all features of the Wolfram language are implemented as valid inputs in WolframAlpha. This is intentional, since if it were possible to evaluate arbitrary Mathematica code in WolframAlpha, it would eliminate much of the incentive to purchase the software.