Let $H$ be the set of third degree polynomials $H = {ax + ax^2 + ax^3 | a in mathbb{C}}$ Is $H$ a subspace of $P_3$? Why or why not? Select all correct answer choices (there may be more than one). a. $H$ is a subspace of $P_3$ because it contains the zero vector of $P_3$ b. $H$ is not a subspace of $P_3$ because it does not contain the zero vector of $P_3$ c. $H$ is not a subspace of $P_3$ because it is not closed under vector addition d. $H$ is a subspace of $P_3$ because it can be written as the span of a subset of $P_3$ e. $H$ is not a subspace of $P_3$ because it is not closed under scalar multiplication f. $H$ is a subspace of $P_3$ because it contains only second degree polynomials g. $H$ is a subspace of $P_3$ because it is closed under vector addition and scalar multiplication
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Does H contain the zero vector of P3? The zero vector of P3 is the polynomial 0x^3 + 0x^2 + 0x + 0. In H, if we set a = 0, we get the polynomial 0x^3 + 0x^2 + 0x, which is the zero vector of P3. So, H contains the zero vector of P3. Show more…
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