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Any other inputs kind help. It will improve my knowledge way of my thinking.

**Another kind help with this question is**

suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the same set V can they both have same bases. (If not kind help with a proof or link of it to improve my knowledge.)

suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the same set V can they both have same bases. (If not kind help with a proof or link of it to improve my knowledge.)

Here O_1,O_2,a_1,a_2 are operations on V.