Let $W$ be a subspace of the space $\mathbb{R}^{4}$ with standart inner product. Let $S=\left\{\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right],\left[\begin{array}{c}-1 \\ 4 \\ 4 \\ 1\end{array}\right]\right\}$ be a basis for $W$. Use the Gram-Schmidt process to transform the basis $S$ into an orthonormal basis.
Answer
\boxed{\text{Final Answer: } \left\{\begin{bmatrix}0.5 \\ 0.5 \\ 0.5 \\ 0.5\end{bmatrix}, \begin{bmatrix}-0.70710678 \\ 0.47140452 \\ 0.47140452 \\ -0.23570226\end{bmatrix}\right\}}
Step 1 ：Apply the Gram-Schmidt process to the given basis S: \(S = \left\{\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\end{bmatrix}, \begin{bmatrix}-1 \\ 4 \\ 4 \\ 1\end{bmatrix}\right\}\)
Step 2 ：Normalize the orthogonal vectors to obtain the orthonormal basis: \(\left\{\begin{bmatrix}0.5 \\ 0.5 \\ 0.5 \\ 0.5\end{bmatrix}, \begin{bmatrix}-0.70710678 \\ 0.47140452 \\ 0.47140452 \\ -0.23570226\end{bmatrix}\right\}\)
Step 3 ：\boxed{\text{Final Answer: } \left\{\begin{bmatrix}0.5 \\ 0.5 \\ 0.5 \\ 0.5\end{bmatrix}, \begin{bmatrix}-0.70710678 \\ 0.47140452 \\ 0.47140452 \\ -0.23570226\end{bmatrix}\right\}}
