If we have $W=\{1-x,1-x+x^2,1+x^2,1-x-x^2\}$ find out of this set forms a basis for $P_2$
I put it into an matrix and row reduce it to get:
$$\begin{bmatrix}1&0&0&2 \cr 0&1&0&-1\cr 0&0&1&0\end{bmatrix}$$
I noticed that while the first three columns are in RRE the fourth column is not and I was wondering:
- Could we just take out the last equation in the set $W$ and that be the basis?
- Is this even considered a basis since we have the last column not in RRE?
- If we are to use the RRE method to determine if this is a basis would we want $0'$s above and below each pivot points since we want to have all of the coefficients be $0$ after doing the RRE?
On the first question: Try it out! Repeat your process and check whether the resulting matrix still has full rank.
On the second question: It is not a basis but a generating set, since a basis is usually defined to be linearly independent. In particular, this fixes how many vectors you have in a basis of some vector space $V$ to be $\operatorname{dim} V$, which is $3$ in your case (I think, anyways).
On the third question: You would want the coefficient matrix to RRE to yield the identity matrix, yes (can you figure out why given the answers to the 2 questions above?)