I have a problem that asks the following:
Determine if the polynomials $p_1(x) = x^3 + x^2 + x + 1$, $p_2(x) = x^2-2x+1$, $p_3(x) = x-1$, and $p_4(x) = x^3-1$ form a basis of $V$ where $V$ is the vector space consisting of all polynomials with real coefficients of degree less or equal to $3$ with usual operations of addition and scalar multiplication of functions.
What I did:
First convert the polynomials to a matrix $$\begin{pmatrix} 1&1&1&1 \\ 0&1&-2&1 \\ 0&0&1&-1 \\ 1&0&0&-1\end{pmatrix}$$
After a series of row reduction
$$\begin{pmatrix} 1&1&1&1 \\ 0&1&-2&1 \\ 0&0&1&-1 \\ 0&0&0&-4\end{pmatrix}$$
I see that the rank of the matrix is $4$, $$rk = 4.$$
I see that the number of vectors is $4$, $$k = 4.$$
I see that $\mathbb{R}^n$ is equal to $\mathbb{R}^4$ and thus $$n = 4.$$
Since $rk = k = n = 4$ then these polynomials indeed form a basis of $V$
How is my solution for this problem? Am I incorrect? If I am, hints leading to the correct way to solve this would be appreciated.