Which of the following set of vectors are bases for P3?

Question

I know that to prove for the bases you have to show 2 things. Linear Independence and Span. Now i have this question below a set of vectors in P3:

    {t^3 + t^2 + t , t^3 - 1 , t^3 + t^2 + t}

The Answers index in the book says they are not the bases for P3 whereas i have worked again and again on this one to make sure that it is linearly independent and it spans P3.Also there is another question(Below) which the book says is not the bases for P3 while infact to me it satisfies the two properties of bases:

      {t^3 - t , t^3 + t^2 + 1 , t - 1}

Now obviously i am doing something wrong but i can't figure it out.Please enlighten me on this one while pretending that i am the dumbest student you have ever seen.

Much obliged !

Answer 1

The first set has two elements which are identical. Ie: it only has 2 elements and thus cannot be LI.