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The problem asks to determine the dimension of the subspace of \(\mathbb{R}^4\) defined by vectors of the form \((0, a, b, 0)\).

### Solution:
The vector \((0, a, b, 0)\) in \(\mathbb{R}^4\) can be written as:
\[
(0, a, b, 0) = a(0, 1, 0, 0) + b(0, 0, 1, 0),
\]
where \(a\) and \(b\) are real numbers.

#### Observations:
1. The vectors \((0, 1, 0, 0)\) and \((0, 0, 1, 0)\) are **linearly independent**.
2. Any vector in this subspace is a linear combination of these two vectors.

#### Conclusion:
Since there are two linearly independent vectors that span the subspace, the **dimension of the subspace** is:
\[
\boxed{2}.
\]

Let me know if you'd like a more detailed breakdown!

### Related Questions:
1. What is the basis for the subspace in this problem?
2. How can we generalize the process of finding the dimension of a subspace?
3. What is the geometric interpretation of this subspace in \(\mathbb{R}^4\)?
4. How does this compare to the full dimension of \(\mathbb{R}^4\)?
5. How do linearly dependent vectors affect the dimension of a subspace?

### Tip:
Always check for linear independence when finding the dimension of a subspace. Linearly independent vectors form the basis of a subspace.

Find the dimension of the following subspace of R^4: All vectors of the form (0, a, b, 0).

Learn how to find the dimension of a subspace of R^4 defined by vectors of the form (0, a, b, 0). This problem explains the concept of linear independence, basis vectors, and dimension calculation step by step, suitable for first-year linear algebra students.

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