Let $\mathbf{W}$ be the subspace spanned by $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$. Write $\mathbf{y}$ as the sum of a vector in $W$ and a vector orthogonal to W \[ \mathbf{y}=\left[\begin{array}{r} 25 \\ 5 \\ 13 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right] \]
Solution
Step 1 ：Let \(\mathbf{W}\) be the subspace spanned by \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). We are given \(\mathbf{y} = \left[\begin{array}{r} 25 \\ 5 \\ 13 \end{array}\right]\), \(\mathbf{u}_{1} = \left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right]\), and \(\mathbf{u}_{2} = \left[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right]\).
Step 2 ：We are asked to express the vector \(\mathbf{y}\) as the sum of a vector in the subspace \(\mathbf{W}\) spanned by vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\), and a vector orthogonal to \(\mathbf{W}\). This can be achieved by first projecting \(\mathbf{y}\) onto the subspace \(\mathbf{W}\), and then subtracting this projection from \(\mathbf{y}\) to get the orthogonal component.
Step 3 ：The projection of \(\mathbf{y}\) onto \(\mathbf{W}\) can be calculated using the formula: \(\text{proj}_\mathbf{W}(\mathbf{y}) = \frac{\mathbf{y} \cdot \mathbf{u}_{1}}{\mathbf{u}_{1} \cdot \mathbf{u}_{1}} \mathbf{u}_{1} + \frac{\mathbf{y} \cdot \mathbf{u}_{2}}{\mathbf{u}_{2} \cdot \mathbf{u}_{2}} \mathbf{u}_{2}\).
Step 4 ：The orthogonal component is then given by: \(\mathbf{y}_{\text{orthogonal}} = \mathbf{y} - \text{proj}_\mathbf{W}(\mathbf{y})\).
Step 5 ：Calculating these, we find \(\text{proj}_\mathbf{W}(\mathbf{y}) = \left[\begin{array}{l} 24 \\ 9 \\ 12 \end{array}\right]\) and \(\mathbf{y}_{\text{orthogonal}} = \left[\begin{array}{l} 1 \\ -4 \\ 1 \end{array}\right]\).
Step 6 ：\(\boxed{\text{Final Answer:}}\) The vector \(\mathbf{y}\) can be written as the sum of the vector \(\left[\begin{array}{l} 24 \\ 9 \\ 12 \end{array}\right]\) in the subspace \(\mathbf{W}\) and the vector \(\left[\begin{array}{l} 1 \\ -4 \\ 1 \end{array}\right]\) orthogonal to \(\mathbf{W}\). So, \(\mathbf{y} = \left[\begin{array}{l} 24 \\ 9 \\ 12 \end{array}\right] + \left[\begin{array}{l} 1 \\ -4 \\ 1 \end{array}\right]\).
