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?

---

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.