--- Commentary: $\textbf{Exercise 6.}$ Prove or give a counterexample: If $p_0, p_1, p_2, p_3$ is a list in $\mathcal{P}_3(\mathbb{F})$ such that none of the polynomials $p_0, p_1, p_2, p_3$ has degree 2, then $p_0, p_1, p_2, p_3$ is not a basis of $\mathcal{P}_3(\mathbb{F})$. $\textbf{Solution 6.}$ To construct a counterexample, define $p_0, p_1, p_2, p_3 \in \mathcal{P}_3(\mathbb{F})$ by \begin{align*} p_0(z) &= 1, \ p_1(z) &= z, \ p_2(z) &= z^2 +z^3, \ p_3(z) &= z^3. \end{align*} None of the polynomials $p_0, p_1, p_2, p_3$ has degree 2, but $p_0, p_1, p_2, p_3$ is a basis of $\mathcal{P}_3(\mathbb{F})$, as is easy to verify. $\textit{Commentary:}$ This exercise asks to either prove or disprove (by counterexample) the claim that if a list of four polynomials in $\mathcal{P}_3(\mathbb{F})$ (the space of polynomials over $\mathbb{F}$ with degree at most 3) doesn't include a polynomial of degree 2, then it cannot be a basis for $\mathcal{P}_3(\mathbb{F})$. The solution provides a counterexample to disprove the claim. The counterexample consists of four polynomials of degrees 0, 1, 3, and 3, respectively. Despite not including a polynomial of degree 2, this list is a basis for $\mathcal{P}_3(\mathbb{F})$. To see why, note that any polynomial in $\mathcal{P}_3(\mathbb{F})$ can be uniquely written as a linear combination of these four polynomials: $a_0 + a_1z + a_2(z^2 + z^3) + a_3z^3 = a_0 + a_1z + a_2z^2 + (a_2 + a_3)z^3.$ This covers all possible polynomials of degree at most 3. Moreover, no non-trivial linear combination of these polynomials is zero, so they are linearly independent. This example illustrates that the condition of having polynomials of all degrees from 0 to 3 is sufficient for a list to be a basis of $\mathcal{P}_3(\mathbb{F})$, but it is not necessary. What matters is that the list spans the entire space and is linearly independent, regardless of the specific degrees of the polynomials. $\textit{Examples:}$ 1. In $\mathcal{P}_4(\mathbb{R})$, the list $1, x, x^3, x^4, x^4 + x^2$ is a basis, despite not including a polynomial of degree 2. 2. In $\mathcal{P}_5(\mathbb{C})$, the list $1, z, z^2, z^4, z^5, z^5 + z^3$ is a basis, despite not including a polynomial of degree 3. 3. In $\mathcal{P}_3(\mathbb{F}_2)$, the list $1, x, x^3, x^3 + x$ is a basis, despite not including a polynomial of degree 2. These examples further demonstrate that a basis for $\mathcal{P}_n(\mathbb{F})$ can take many different forms. The key is that the polynomials collectively span the entire space and are linearly independent, not that they include polynomials of every degree up to $n$. ---