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Let's go over each exercise from your homework on linear algebra.

### Exercise 1
You are given three vectors:
- \( v_1 = \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix} \)
- \( v_2 = \begin{pmatrix} 1 \\ 2 \\ 6 \end{pmatrix} \)
- \( v_3 = \begin{pmatrix} 0 \\ 4 \\ 3 \end{pmatrix} \)

These vectors belong to \( \mathbb{R}^3 \).

1. **Part (a)**: Show that \( \mathcal{B} = (v_1, v_2, v_3) \) is a basis of \( \mathbb{R}^3 \).
   - To show that \( \mathcal{B} \) is a basis, you need to demonstrate that \( v_1, v_2, \) and \( v_3 \) are linearly independent and that they span \( \mathbb{R}^3 \).
   - To check for linear independence, set up the equation \( c_1 v_1 + c_2 v_2 + c_3 v_3 = 0 \) and see if the only solution is \( c_1 = c_2 = c_3 = 0 \).
   - You can also confirm that the vectors form a basis by checking that the matrix formed by placing \( v_1, v_2, \) and \( v_3 \) as columns has a non-zero determinant.

2. **Part (b)**: Express each vector of the standard basis \( \mathcal{E} = (e_1, e_2, e_3) \), where
   - \( e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \),
   - \( e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \),
   - \( e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \),
   
   as linear combinations of \( \mathcal{B} \).
   - This requires you to find scalars \( a_1, a_2, a_3 \) such that \( e_i = a_1 v_1 + a_2 v_2 + a_3 v_3 \) for each \( i = 1, 2, 3 \).
   - Solve these systems of equations to express each standard basis vector in terms of \( v_1, v_2, \) and \( v_3 \).

### Exercise 2
Construct a set \( X \subseteq \mathbb{R}^2 \) satisfying the following conditions:
- \( X \neq \emptyset \)
- \( X \) is closed under scalar multiplication.
- \( X \) is *not* a subspace of \( \mathbb{R}^2 \).

If you believe such a construction is not possible, you’ll need to explain why.

   - Recall that a subspace of \( \mathbb{R}^2 \) must satisfy three conditions: it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.
   - Think about subsets that are closed under scalar multiplication but fail other subspace criteria.

Would you like a detailed solution or further assistance with any specific part? 

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### Follow-up Questions:
1. What techniques are used to determine if vectors are linearly independent?
2. How can determinants help verify if vectors form a basis?
3. What properties distinguish subspaces from other sets in vector spaces?
4. Can a set be closed under scalar multiplication but not vector addition?
5. How does expressing vectors as linear combinations relate to changing bases?

#### Tip:
Remember that showing linear independence often simplifies with matrix methods, like row reduction or determinant checks.

Let's go over each exercise from your homework on linear algebra. Exercise 1: You are given three vectors: v1 = (1, 1, 5), v2 = (1, 2, 6), v3 = (0, 4, 3). Show that the set B = (v1, v2, v3) is a basis of R^3. Exercise 2: Construct a set X ⊆ R^2 satisfying the conditions that X is closed under scalar multiplication but is not a subspace of R^2.

Solve problems involving vector spaces, linear independence, and subspaces in this detailed linear algebra homework. Learn how to show a set of vectors forms a basis of R^3, express vectors as linear combinations, and construct a set closed under scalar multiplication but not a subspace of R^2.

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