(Solved): Exercise 1 Orthogonal Projections (3+3+3+4+6+3credits) Consider the Euclidean vector space R3 w ...

Exercise 1 Orthogonal Projections $(3+3+3+4+6+3\text{ credits })$ Consider the Euclidean vector space ${\mathbb{R}}^3$ with the dot product. A subspace $U?{\mathbb{R}}^3$ and vector $\mathbf{x}?{\mathbb{R}}^3$ are given by: $U=\mathrm{span}??\begin{array}{c}?1\\ 1\\ 1\end{array}?,?\begin{array}{c}2\\ ?1\\ ?2\end{array}??,\mathbf{x}=?\begin{array}{c}8\\ 4\\ 16\end{array}?$ 1. Show that $\mathbf{x}?/U$. 2. Determine the orthogonal projection of $\mathbf{x}$ onto $U$, denoted ${?}_U(\mathbf{x})$. 3. Determine the distance $d(\mathbf{x},U):={\min }_{\mathbf{y}?U}?\mathbf{x}?\mathbf{y}?$, where $???$ denotes the Euclidean norm. 4. Use Gram-Schmidt orthogonalization to transform the matrix $\mathbf{A}=?\begin{array}{cc}?1& 2\\ 1& ?1\\ 1& ?2\end{array}?$ into a matrix $\mathbf{B}$ with orthonormal columns. 5. Let $\mathbf{Q}?{\mathbb{R}}^{m\times n}$ be a matrix with orthonormal columns and $\mathbf{x}?{\mathbb{R}}^m$ be an m-dimensional vector. Find the vector $?$ that minimizes $?\mathbf{x}?\mathbf{Q}?{?}^2+???{?}^2$, where $?$ is a positive real number. 6. Compute the vector $?$ for the matrix $\mathbf{B}$ and $?=10$.