1. **State the problem:** We have a subspace $V = \text{L}\{(1,1,k)\}$ where $k$ is any real number. We want to find the dimension of $V$. 2. **Understand the subspace:** The set $V$ consists of all vectors of the form $(1,1,k)$ for any real number $k$. This means $V$ is the span of the vector $(1,1,k)$ where $k$ varies. 3. **Check linear independence:** Since $k$ can be any real number, the vectors $(1,1,k)$ form a line in $\mathbb{R}^3$ parameterized by $k$. However, the vector $(1,1,k)$ can be written as $(1,1,0) + k(0,0,1)$. 4. **Rewrite the subspace:** So $V = \text{span}\{(1,1,0), (0,0,1)\}$. 5. **Check if these two vectors are linearly independent:** - $(1,1,0)$ is not a scalar multiple of $(0,0,1)$. - Therefore, they are linearly independent. 6. **Dimension:** Since $V$ is spanned by two linearly independent vectors, the dimension of $V$ is 2. **Final answer:** $$\boxed{\dim(V) = 2}$$