# Finding a basis for subspace of $\mathbb{P_{3}}$

Here's the question: find a basis for the subspace of $$\mathbb{P_{3}}$$ consisting of all vectors of the form $$at^3-bt^2+ct+d$$, where $$c=a-2d$$ and $$b=5a+3d$$. And here's my solution: First of all, I put the values that mentioned in the question and I got: $$at^3-(5a+3d)t^2+(a-2d)t+d=a(t^3-5t^2+t)+d(-3t^2-2t+1)$$ So, it can be seen that it spans, say $$\mathbb{W}$$. More rigorously $$\{t^3-5t^2+t,-3t^2-2t+1\}$$ spans $$\mathbb{W}$$. Now, we must check whether it is linearly independent. In order to do that we must attain some scalars: $$t^3(a_{1})+t^2(-5a_{1}-3a_{2})+t(a_{1}-a_{2})+a_{2}=0$$. One can easily see that $$a_{1}=a_{2}=0$$. Therefore $$\{t^3-5t^2+t,-3t^2-2t+1\}$$ is a basis for $$\mathbb{W}$$ Do I make any make? Are there any gaps? Thanks for any help.

• Looks fine to me. – Andrei Jan 13 at 20:27
• Okay thank you so much, it helped! – beingmathematician Jan 13 at 20:29