Orthogonal basis with respect to the inner product

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I have been trying to see if only two of the vectors are basis for the span, so that I don't have to do gram schmidt process for all three vectors because it gets very complicated with all the t's.

When t = 1, only u1 is a basis for the span, but for all other values of t, I think u1,u2 and u3 are basis for the span.

Does anyone have any recommendations?