Let $P_3$ be the set of all real polynomials of degree 3 or less. This set forms a real vector space. Show that $\{2x^3+x+1,x−2,x^3−x^2\}$ is a linearly independent set, and find a basis for $P_3$ which includes these three polynomials.
Linear independence is easy, I just put the coefficients in a set and rref;
$\begin{bmatrix}2&1&1\\0&1&-2\\1&-1&0\end{bmatrix} => rref =>\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
So I've proven that the set is linearly independent. Also I have 3 pivots for 3 variables, so think I can write the vector space $P_3$ as a span of the polynomials (or, vectors of the coefficients);
$P_3 = span\{(2, 1, 1),(0, 1,-2),(1, -1, 0)\}$
Since a basis is simply a set of linearly independent vectors which cover the span, are these three vectors a basis for $P_3$, or do I have more work to do? (EDIT: That is, if we pretend that I hadn't completely erased an entire degree from the polynomials).
Also, given that $P_3$ is talked about in terms of polynomials (and assuming I've done the above properly); should I answer questions like this using the polynomials themselves, or is it okay to simply use vectors/matrices/etc with the coefficients?
Best Answer
Hint: how would one write the the polynomial of degree $n<1$ in your space?
The polynomial of degree $0$ is of the form $c \ x^0$. Meaning it's just a constant. This is how your matrix should look like initially. From top to bottom coefficients for $x^3,x^2,x^1,x^0$. Your vectors should have 4 components not 3. \begin{bmatrix} 2&0 &1 \\ 0&0 &-1 \\ 1&1&0 \\ 1&-2 &0 \end{bmatrix}
Oh answering your question about the polynomials or the matrices. We use the matrices because it's easier to show linear independence with them. Showing independence is equivalent to showing $A\vec x=\vec 0$ $\Rightarrow \vec x=\vec 0$. It would be hard to use polynomials themselves to show that.
The columns of the following matrix would form a basis since they are linearly independent. \begin{bmatrix} 2&0 &1 &1 \\ 0&0 &-1& 0\\ 1&1&0 & 0\\ 1&-2 &0 &0 \end{bmatrix}