The image contains a set of mathematical problems related to vector spaces, subspaces, norms, and matrices. I'll go through each question based on the sections visible in the image:

Section I:

1. Problem 1(a): Question: Examine whether $V$ defined is a vector space or not. If $V$ is not a vector space, then state which of the properties are not satisfied.
   Details:

   • $V = \{(x, y, z) \mid x, y, z \in \mathbb{R} \}$ with vector addition and scalar multiplication defined as:
     $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (3x_1 + 4u, y_1 - 2v, z_1 + w)$ $\alpha \cdot (x, y, z) = (\alpha x, \alpha y, \alpha z)$
   • You need to check whether this structure satisfies the axioms of a vector space.

2. Problem 1(b): Question: Let $V_3(F)$ be the set of all 3x1 real matrices. Show that the set $S = \left\{ \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ -\frac{1}{2} \end{pmatrix} \right\}$ forms a basis of $V_3(F)$.

Section II:

3. Problem 2(a): Question:
   Let $V$ be the vector space of all triplets of the form $(x_1, x_2, x_3)$ in $\mathbb{R}^3$ with the usual addition and scalar multiplication.
   Define $W = \{(x_1, x_2, x_3) \mid x_1, x_2, x_3 \in \mathbb{R}, x_1^2 + x_2^2 + x_3^2 \leq 4\}$.
   Is $W$ a subspace of $V$? If not, state why.

4. Problem 2(b): Question: Define a normed vector space and distance between two vectors with respect to inner product space. State their properties.

Would you like detailed solutions to any of these problems or explanations for any concepts?

Here are some relative questions to deepen understanding:

1. What are the key axioms that must be satisfied for a set to be a vector space?
2. How can you prove that a set of vectors forms a basis for a vector space?
3. Why is the set $W$ in problem 2(a) not necessarily a subspace of $V$?
4. How do norms and inner products relate to the concept of distance between vectors?
5. What is the geometric significance of a subspace within a vector space?

Tip: When proving whether a subset is a subspace, always check the three subspace criteria: closure under addition, closure under scalar multiplication, and whether it contains the zero vector.