(Solved): please will upvote roblem 4: Let W be the subspace of \( R^{1} \) spanned by vectars \( \overrightar ...

roblem 4: Let W be the subspace of \( R^{1} \) spanned by vectars \( \overrightarrow{\mathrm{u}}_{1}=\left[\begin{array}{r}1 \\ 0 \\ -1\end{array}\right] \) and \( \overrightarrow{\mathrm{u}}_{2}=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right] \). It's the plane \( z=-x \). (9 point The vectors \( \overrightarrow{\mathrm{u}}_{1} \) and \( \overrightarrow{\mathrm{u}}_{2} \) are an orthonormal basis for the plane \( z=-x \). i. True Find the 2-nom of the new vector \( \vec{y}=\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right] \) i. \( \sqrt{3} \quad \) ill \( \quad \) iil. \( 5.9 \). Orthogonal Decomposition: Using the same vector \( \vec{y}=\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right] \) as above, find vectors \( \hat{y} \) and \( \vec{z} \) such that \( \vec{y}=\hat{y}+\overrightarrow{1} \), where \( \hat{y} \) is the orthogonal projection of \( \vec{y} \) onto the plane \( W \), and \( \vec{z} \) is a vector crthogonal to \( W \). i. First find \( \hat{y} \), the orthogonal projection of \( \vec{y} \) onto the plane \( W \). Record \( \hat{y} \) here. Hint: \( \hat{y}=\frac{y \vec{u}_{1}}{\vec{u}_{1} \vec{u}_{1}} \vec{u}_{1}+\frac{\vec{y} \vec{u}_{2}}{\vec{u}_{2} \vec{u}_{2}} \vec{u}_{2} \) ii. Now find \( \vec{z} \), the componant orthogomal to \( W \). Record \( z \) bere. d. Verify your results by proving vector \( \bar{y} \) and \( \bar{z} \) are orthegonal to each other: Show all details. e. Now let's eppand \( \vec{u}_{2} \) and \( \vec{u}_{2} \) to a basir by adding the third vector \( \vec{u}_{2}=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right] \). Record \( U^{\tau} U \) here. Fotm the matrix \( v=\left[\overrightarrow{\mathrm{u}}_{2} \quad \overrightarrow{\mathrm{u}}_{2} \quad \overrightarrow{\mathrm{u}}_{1}\right] \) and compute \( U^{\tau} U \). Show all detore L The invase of \( U \) is \( U^{\top} \). True or false? Find the coefficient: \( c_{1} \) wo that \( \vec{y}=c_{1} \vec{v}_{i}+c_{2} \vec{v}_{2}+c_{2} \vec{u} \), and collact into a column vector. Mint: \( c_{2}=\frac{\sum \bar{y} \vec{u}_{1}}{a_{1}+u_{i}} \)