The problem asks for the vector $(a, b)$ in the subspace $U \subset \mathbb{R}^2$, which is spanned by the vector $(1, 2)$, such that the difference $(a, b) - (5, 8)$ is minimized.

Step 1: Subspace Definition

The subspace $U$ is the set of all scalar multiples of the vector $(1, 2)$. Therefore, any vector in $U$ can be written as:

$(a, b) = t(1, 2) = (t, 2t)$

where $t$ is a scalar.

Step 2: Minimizing the Difference

We need to minimize the distance between the vector $(a, b)$ and the vector $(5, 8)$. This distance is the magnitude of the difference:

$\text{distance} = \| (a, b) - (5, 8) \| = \| (t - 5, 2t - 8) \|$

The goal is to minimize this distance. To do so, we can use the projection of the vector $(5, 8)$ onto the subspace $U$. The vector in $U$ closest to $(5, 8)$ is the projection of $(5, 8)$ onto the span of $(1, 2)$.

Step 3: Projection Formula

The formula for the projection of a vector $\mathbf{v} = (x, y)$ onto a vector $\mathbf{u} = (u_1, u_2)$ is given by:

$\text{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$

For the vectors $\mathbf{v} = (5, 8)$ and $\mathbf{u} = (1, 2)$, we compute:

$\mathbf{v} \cdot \mathbf{u} = 5 \cdot 1 + 8 \cdot 2 = 5 + 16 = 21$

$\mathbf{u} \cdot \mathbf{u} = 1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5$

Thus, the projection is:

$\text{proj}_{(1, 2)}(5, 8) = \frac{21}{5} (1, 2) = \left(\frac{21}{5}, \frac{42}{5}\right)$

Step 4: The Values of $a$ and $b$

The vector in $U$ closest to $(5, 8)$ is $\left(\frac{21}{5}, \frac{42}{5}\right)$. Therefore, the value of $a$ is:

$a = \frac{21}{5} = 4.2$

Final Answer:

$a = 4.2$

Would you like more details or have any other questions? Here are some related questions:

1. How do we calculate the distance between two vectors?
2. What is the concept of a subspace in linear algebra?
3. How does vector projection work geometrically?
4. What is the significance of the dot product in finding projections?
5. Can the projection formula be used for higher-dimensional subspaces?

Tip: When working with projections, it’s useful to visualize them as the "shadow" of one vector onto another, which gives the closest vector in the subspace.