To determine if $U$ is a subspace of $V$ in each part of the question, we need to verify that $U$ meets the criteria for a subspace of $V$. Specifically, $U$ should satisfy three main conditions:

1. Contain the zero vector of $V$.
2. Be closed under vector addition.
3. Be closed under scalar multiplication.

Let's analyze each part:

Part (a)

• Given: $V = \mathbb{R}^3$ and $U = \{(x, y, z) \in \mathbb{R}^3 \mid x + 7y + 5z = 3\}$.
• Analysis:
  • This is a plane in $\mathbb{R}^3$, but it does not contain the zero vector (since $0 + 7\cdot0 + 5\cdot0 = 0 \neq 3$).
  • Therefore, $U$ is not a subspace of $V$.

Part (b)

• Given: $V = P_3(\mathbb{R})$ (polynomials of degree at most 3) and $U = \{p(t) \in P_3(\mathbb{R}) \mid p(t) = at + bt^2, \text{ where } a, b \in \mathbb{R} \}$.
• Analysis:
  • $U$ consists of polynomials of degree at most 2. It includes the zero polynomial (by setting $a = 0$ and $b = 0$).
  • $U$ is closed under addition and scalar multiplication.
  • Therefore, $U$ is a subspace of $V$.

Part (c)

• Given: $V$ is the set of all $2 \times 2$ matrices with real entries, and $U$ is the set of matrices of the form $\begin{pmatrix} 0 & a \\ b & 0 \end{pmatrix}$, where $a, b \in \mathbb{R}$.
• Analysis:
  • $U$ includes the zero matrix (by setting $a = 0$ and $b = 0$).
  • $U$ is closed under addition and scalar multiplication.
  • Therefore, $U$ is a subspace of $V$.

Would you like further details on any of these steps, or do you have specific questions?

Here are some related questions to explore:

1. How can we confirm closure properties for subspaces more rigorously?
2. What makes a set of polynomials a subspace in general?
3. Why does the condition of containing the zero vector matter for subspaces?
4. How does the structure of a matrix affect whether it forms a subspace?
5. Can we visualize subspaces in $\mathbb{R}^3$ geometrically?

Tip: Remember, a non-homogeneous equation (like $x + 7y + 5z = 3$) cannot form a subspace, as it doesn’t include the origin.