We are given the subspace $W = \{(x_1, x_2, x_3, x_4, x_5) \in \mathbb{R}^5 \mid 10x_1 + 6x_5 = 0, 10x_2 + 10x_4 = 0\}$, which is defined by two linear equations:

1. $10x_1 + 6x_5 = 0$
2. $10x_2 + 10x_4 = 0$

Step 1: Solve the system of equations

Let's analyze these constraints:

From the first equation $10x_1 + 6x_5 = 0$:

$x_1 = -\frac{3}{5} x_5$ Thus, $x_1$ depends on $x_5$.

From the second equation $10x_2 + 10x_4 = 0$:

$x_2 = -x_4$ So, $x_2$ depends on $x_4$.

Step 2: Express the general form of the vectors in $W$

Now, we can express a vector in $W$ as:

$(x_1, x_2, x_3, x_4, x_5) = \left(-\frac{3}{5} x_5, -x_4, x_3, x_4, x_5 \right)$

Here, the components $x_3$, $x_4$, and $x_5$ are free variables, so we can represent any vector in $W$ as a linear combination of three independent vectors corresponding to these free variables:

$\mathbf{v}(x_3, x_4, x_5) = x_3 \left( 0, -1, 1, 0, 0 \right) + x_4 \left( -\frac{3}{5}, -1, 0, 1, 0 \right) + x_5 \left( -\frac{3}{5}, 0, 0, 0, 1 \right)$

Thus, we have three free variables $x_3$, $x_4$, and $x_5$, implying that the subspace $W$ is spanned by three independent vectors. Therefore, the dimension of $W$ is 3.

Conclusion:

The dimension of $W$ is $\boxed{3}$.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the rank of the matrix that represents the system of equations for $W$?
2. How do the constraints affect the nullity of a matrix?
3. What is the geometric interpretation of a subspace defined by linear equations?
4. How can you check if a set of vectors is linearly independent?
5. What would happen if the equations defining $W$ were different?

Tip: When working with subspaces defined by linear equations, always check the number of free variables to determine the dimension.