Question

Let $\mathbb{P}_3$ be the real vector space consisting of polynomials of degree at most 3. An element of $\mathbb{P}_3$ can be written $p(t) = a_0 + a_1t + a_2t^2 + a_3t^3$, with $a_0, a_1, a_2, a_3 \in \mathbb{R}$. (a) Show that $U := \{p(t) \in \mathbb{P}_3 \mid a_0 + a_2 = 0\}$ is a subspace of $\mathbb{P}_3$. (b) Show that the vectors $1 + t$, $t + t^2$, $t^2 + t^3$, $t^3$ are linearly independent in $\mathbb{P}_3$.

          Let $\mathbb{P}_3$ be the real vector space consisting of polynomials of degree at most 3. An element of $\mathbb{P}_3$ can be written $p(t) = a_0 + a_1t + a_2t^2 + a_3t^3$, with $a_0, a_1, a_2, a_3 \in \mathbb{R}$.
(a) Show that $U := \{p(t) \in \mathbb{P}_3 \mid a_0 + a_2 = 0\}$ is a subspace of $\mathbb{P}_3$.
(b) Show that the vectors $1 + t$, $t + t^2$, $t^2 + t^3$, $t^3$ are linearly independent in $\mathbb{P}_3$.

$Let ℙ3 be the real vector space consisting of polynomials of degree at most 3. An element of ℙ3 can be written p(t) = a0 + a1t + a2t^2 + a3t^3, with a0, a1, a2, a3 ∈ℝ. (a) Show that U := {p(t) ∈ℙ3 | a0 + a2 = 0} is a subspace of ℙ3. (b) Show that the vectors 1 + t, t + t^2, t^2 + t^3, t^3 are linearly independent in ℙ3.$

Added by Juan Carlos M.

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