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The problem asks to find the values of \( a \in \mathbb{R} \) such that the vector:

\[
\vec{v} = (2a + 12, -1 + 2a, -2)
\]

belongs to the subspace \( S \subset \mathbb{R}^3 \), where the basis for the subspace \( S \) is given by:

\[
\{(-1, -1, 0), (1, 5, -1)\}.
\]

To solve this problem, we need to express the vector \( \vec{v} \) as a linear combination of the basis vectors \( (-1, -1, 0) \) and \( (1, 5, -1) \). This means we need to find scalars \( \lambda_1 \) and \( \lambda_2 \) such that:

\[
\vec{v} = \lambda_1 (-1, -1, 0) + \lambda_2 (1, 5, -1).
\]

This gives the following system of equations by comparing the components of the vectors:

1. \( 2a + 12 = -\lambda_1 + \lambda_2 \)
2. \( -1 + 2a = -\lambda_1 + 5\lambda_2 \)
3. \( -2 = 0\lambda_1 - \lambda_2 \)

From the third equation, we can solve for \( \lambda_2 \):

\[
-2 = -\lambda_2 \implies \lambda_2 = 2.
\]

Substituting \( \lambda_2 = 2 \) into the first two equations:

1. \( 2a + 12 = -\lambda_1 + 2 \)
2. \( -1 + 2a = -\lambda_1 + 10 \)

Now solve these two equations:

1. \( 2a + 12 = -\lambda_1 + 2 \implies -\lambda_1 = 2a + 10 \implies \lambda_1 = -(2a + 10) \)
2. \( -1 + 2a = -(2a + 10) + 10 \implies -1 + 2a = -2a - 10 + 10 \implies -1 + 2a = -2a \)

Solve the second equation for \( a \):

\[
-1 + 2a = -2a \implies 4a = 1 \implies a = \frac{1}{4}.
\]

Thus, the value of \( a \) for which the vector \( \vec{v} \) belongs to the subspace \( S \) is:

\[
a = \frac{1}{4}.
\]

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to explore this concept further:

1. How can you generalize the process of determining whether a vector belongs to a subspace?
2. What happens to the solution if the vector does not belong to the subspace?
3. Can a vector belong to a subspace with a different dimension than its own space?
4. How does the choice of basis vectors affect the process of finding the scalars \( \lambda_1 \) and \( \lambda_2 \)?
5. What are the geometric interpretations of subspaces in \( \mathbb{R}^3 \)?

**Tip:** When solving systems involving vector spaces, it's useful to check the rank of the coefficient matrix to ensure the system has a solution.

Find all values of a ∈ R such that the vector v = (2a + 12, -1 + 2a, -2) belongs to the subspace S ⊆ R^3 with basis {(-1, -1, 0), (1, 5, -1)}.

Discover how to find the values of 'a' that make the vector (2a + 12, -1 + 2a, -2) belong to a specific subspace in R^3 with a given basis. This problem involves concepts from linear algebra such as vector spaces, linear combinations, and solving systems of linear equations.

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