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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.

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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.

Examine whether V is a vector space, prove whether a set forms a basis, investigate subspaces and normed vector spaces, and classify matrices based on their definiteness.

Explore solutions to vector space and matrix algebra problems, including examining vector spaces, proving sets form a basis, and classifying matrices based on their definiteness. Detailed explanations for undergraduate-level students.

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