To solve the problem of determining which pair of vectors is linearly dependent, let's first recall that two vectors $\mathbf{u}_1$ and $\mathbf{u}_2$ in $\mathbb{R}^2$ are linearly dependent if there exists a scalar $\lambda$ such that:

$\mathbf{u}_2 = \lambda \mathbf{u}_1$

This means that one vector is a scalar multiple of the other. If no such $\lambda$ exists, the vectors are linearly independent.

Now, let's analyze each pair of vectors.

1. First Option:

$\mathbf{u}_1 = (1, 2), \quad \mathbf{u}_2 = (5, 10)$

Check if there is a scalar $\lambda$ such that:

$\mathbf{u}_2 = \lambda \mathbf{u}_1$

This gives the system of equations: $5 = \lambda \cdot 1$ $10 = \lambda \cdot 2$

From the first equation, $\lambda = 5$. Substituting into the second equation:

$10 = 5 \cdot 2 = 10$

Both equations are satisfied, so $\mathbf{u}_1$ and $\mathbf{u}_2$ are linearly dependent.

2. Second Option:

$\mathbf{u}_1 = (2, 8), \quad \mathbf{u}_2 = (2, 5)$

Again, check if there is a scalar $\lambda$ such that:

$\mathbf{u}_2 = \lambda \mathbf{u}_1$

This gives the system of equations: $2 = \lambda \cdot 2$ $5 = \lambda \cdot 8$

From the first equation, $\lambda = 1$. Substituting into the second equation:

$5 = 1 \cdot 8 = 8$

This is not true, so $\mathbf{u}_1$ and $\mathbf{u}_2$ are linearly independent.

3. Third Option:

$\mathbf{u}_1 = (1, 2), \quad \mathbf{u}_2 = (2, 5)$

Check if there is a scalar $\lambda$ such that:

$\mathbf{u}_2 = \lambda \mathbf{u}_1$

This gives the system of equations: $2 = \lambda \cdot 1$ $5 = \lambda \cdot 2$

From the first equation, $\lambda = 2$. Substituting into the second equation:

$5 = 2 \cdot 2 = 4$

This is not true, so $\mathbf{u}_1$ and $\mathbf{u}_2$ are linearly independent.

Conclusion:

The first pair of vectors $\mathbf{u}_1 = (1, 2)$ and $\mathbf{u}_2 = (5, 10)$ is linearly dependent. Therefore, the correct answer is the first option.

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the definition of linear independence in higher dimensions?
2. How do we check linear dependence using matrix echelon form?
3. What happens if two vectors are linearly dependent geometrically?
4. Can linearly dependent vectors form a basis for $\mathbb{R}^2$?
5. What is the importance of linear independence in solving systems of equations?

Tip: If two vectors in $\mathbb{R}^2$ are linearly dependent, their cross product will always be zero, as they lie on the same line.