The problem requires us to find the vector u in the subspace $U$ that is closest to a given vector $\mathbf{x} = (3, 0, 1, 1)$ and express $\mathbf{x}$ as the sum of a vector $\mathbf{u} \in U$ and a vector $\mathbf{v}$, where $\mathbf{v}$ is orthogonal to $U$. Here's how we can solve this:

Step-by-Step Approach

1. Identify the subspace $U$:
   The subspace $U$ is the span of two vectors in $\mathbb{R}^4$, given by: $\mathbf{x_1} = (3, 1, 0, -1), \quad \mathbf{x_2} = (0, 3, 2, 1)$ So, the subspace $U = \text{span}\{\mathbf{x_1}, \mathbf{x_2}\}$.

2. Projection of $\mathbf{x}$ onto $U$:
   To find the vector $\mathbf{u}$ in $U$ closest to $\mathbf{x}$, we need to project $\mathbf{x}$ onto the subspace spanned by $\mathbf{x_1}$ and $\mathbf{x_2}$.

   The projection of $\mathbf{x}$ onto $U$ is given by: $\mathbf{u} = \text{Proj}_U(\mathbf{x}) = c_1 \mathbf{x_1} + c_2 \mathbf{x_2}$ where $c_1$ and $c_2$ are scalars that need to be determined.

3. Find the coefficients $c_1$ and $c_2$: These coefficients can be found by solving the system of linear equations that comes from the condition that $\mathbf{u}$ must minimize the distance between $\mathbf{x}$ and $U$, which is equivalent to the orthogonal projection.

   To find $c_1$ and $c_2$, we solve the equation: $\mathbf{A} \mathbf{c} = \mathbf{b}$ where $\mathbf{A}$ is the matrix formed by $\mathbf{x_1}$ and $\mathbf{x_2}$, $\mathbf{c}$ is the vector of coefficients $(c_1, c_2)$, and $\mathbf{b} = \mathbf{x}$.

   Let me calculate this next.The vector u (the projection of $\mathbf{x}$ onto $U$) and the orthogonal vector v are calculated as follows:

• u = $\left( \frac{53}{25}, \frac{157}{150}, \frac{34}{150}, \frac{-89}{150} \right)$
• v = $\left( \frac{22}{25}, \frac{-157}{150}, \frac{116}{150}, \frac{239}{150} \right)$

Thus, the vector $\mathbf{x} = \mathbf{u} + \mathbf{v}$ can be expressed as:

$\mathbf{x} = \left( \frac{53}{25}, \frac{157}{150}, \frac{34}{150}, \frac{-89}{150} \right) + \left( \frac{22}{25}, \frac{-157}{150}, \frac{116}{150}, \frac{239}{150} \right)$

Would you like further explanation on any part? Here are 5 related questions to explore more:

1. How do we find the projection of a vector onto a subspace?
2. Why does the orthogonal vector $\mathbf{v}$ satisfy $\mathbf{v} \perp U$?
3. What are the properties of the least squares solution in this context?
4. How can this problem be solved using Gram-Schmidt orthogonalization?
5. How does the formula for projection work when there are more than two basis vectors?

Tip: Always check that the orthogonal vector is indeed perpendicular to all basis vectors of the subspace.