To check for linear independence or dependence, we will look to see if there are polynomials in our set that are scalar multiples of each other or if they are linear combinations of each other. Here, we can see immediately that the first polynomial in the set that is t^3+t^2-2t+1 is the sum of the other polynomials in the set that are t^2+1 and t^3-2t. Furthermore, we can see that the last polynomial in the set that is 2t^3+3t^2-4t+3 is just 2(t_3-2t) added with 3(t^2+1). Therefore, since our set of polynomials contain scalar multiples and added polynomials already in our set, we conclude that the set S is linearly independent.
Moving on, we will find a basis for W, To find the basis for W, we will pare down the set S such that it contains linearly independent vectors that span which is the definition of a basis. To pare down the set to form a basis for P3, we will remove the polynomials t^3+t^2-2t+1 and 2t^3+3t^2-4t+3 since they are just linear combinations of the second and third polynomial in our set. When we remove these vectors, our new set becomes {t^2+1, t^3-2t}. These vectors are linearly independent. Now we know that these vectors also span the subspace because we can use these linearly independent vectors to construct any polynomial of P3. Therefore these vectors form a basis.
To find the dimension of the basis, we know that the dimension is defined as the number of linearly independent vectors in our basis or in other words, the number of basis vectors. Here we see that we have two vectors in our basis therefore the dimension of our basis for W is 2.
