The problem asks to compute the projection of the vector $v = (3, 2, 4, -2)$ onto the subspace $W$ spanned by $u_1 = (1, 1, 1, 1)$ and $u_2 = (1, 1, 1, -1)$.

Solution:

The projection of a vector $v$ onto a subspace $W$ spanned by vectors $u_1, u_2, \dots, u_k$ can be computed using the Gram-Schmidt procedure or directly using the projection formula:

$\text{proj}_W(v) = \sum_{i=1}^k \frac{\langle v, u_i \rangle}{\langle u_i, u_i \rangle} u_i$

Here, $\langle \cdot, \cdot \rangle$ denotes the dot product.

1. Step 1: Compute dot products of $v$ with $u_1$ and $u_2$:

   • $\langle v, u_1 \rangle = 3 \cdot 1 + 2 \cdot 1 + 4 \cdot 1 + (-2) \cdot 1 = 3 + 2 + 4 - 2 = 7$
   • $\langle v, u_2 \rangle = 3 \cdot 1 + 2 \cdot 1 + 4 \cdot 1 + (-2) \cdot (-1) = 3 + 2 + 4 + 2 = 11$

2. Step 2: Compute the magnitudes squared of $u_1$ and $u_2$:

   • $\langle u_1, u_1 \rangle = 1^2 + 1^2 + 1^2 + 1^2 = 4$
   • $\langle u_2, u_2 \rangle = 1^2 + 1^2 + 1^2 + (-1)^2 = 4$

3. Step 3: Compute the projections of $v$ onto $u_1$ and $u_2$:

   • $\text{proj}_{u_1}(v) = \frac{\langle v, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1 = \frac{7}{4} (1, 1, 1, 1) = \left( \frac{7}{4}, \frac{7}{4}, \frac{7}{4}, \frac{7}{4} \right)$
   • $\text{proj}_{u_2}(v) = \frac{\langle v, u_2 \rangle}{\langle u_2, u_2 \rangle} u_2 = \frac{11}{4} (1, 1, 1, -1) = \left( \frac{11}{4}, \frac{11}{4}, \frac{11}{4}, -\frac{11}{4} \right)$

4. Step 4: Add the two projections to find $\text{proj}_W(v)$:

   $\text{proj}_W(v) = \text{proj}_{u_1}(v) + \text{proj}_{u_2}(v)$

   Add the components: $\text{proj}_W(v) = \left( \frac{7}{4} + \frac{11}{4}, \frac{7}{4} + \frac{11}{4}, \frac{7}{4} + \frac{11}{4}, \frac{7}{4} - \frac{11}{4} \right)$

   Simplify: $\text{proj}_W(v) = \left( \frac{18}{4}, \frac{18}{4}, \frac{18}{4}, \frac{-4}{4} \right) = \left( \frac{9}{2}, \frac{9}{2}, \frac{9}{2}, -1 \right)$

Final Answer:

The projection of $v$ onto $W$ is: $\text{proj}_W(v) = \left( \frac{9}{2}, \frac{9}{2}, \frac{9}{2}, -1 \right)$

Would you like further clarification or a deeper explanation of any step?

Related Questions:

1. What is the geometric interpretation of projecting $v$ onto $W$?
2. How does the Gram-Schmidt process relate to projections?
3. Can you compute the orthogonal complement of $v$ with respect to $W$?
4. What are the properties of projections onto subspaces?
5. How would the process change if $W$ were spanned by three vectors?

Tip:

Always check if the spanning vectors $u_1, u_2, \dots$ are orthogonal, as this simplifies the projection calculation significantly!