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^{5}

_{3}(The set of elements in the field of modulus 3)

Attemept: So the issue isn't in finding a basis per say. If this was the field of Real numbers I wouldn't have an issue, I would just row reduce and use the corresponding vectors with leading one's. But my issue is with this field. When I row reduce I end up in a situation where the only way I could get leading ones is by using a fraction, but if this is the set of modulus 3, then isn't it only defined for the Natural numbers? i.e: fractions don't exist in this field? So how can I find the basis then?