I have a linear algebra question I need to translate from German to English. Here is the question. All your help is much appreciated:
Here is the google translated piece: Problem 2 (20 points) We consider Rn with the standard scalar product. Let U⊆ Rn is an r-dimensional (0 ≤ r ≤ n) linear subspace with orthonormal basis B = {u1,. . . , ur}. Let PU : Rn → Rn, where the orthogonal projection onto U, defined by PU(v) =j=1r(uj,v) uj
(A) Show that PU is self-adjoint. (B) Denote by U the orthogonal complement to U (see Series 9, Task 2). Verify for all v ∈ Rn the Pythagorean theorem, ie the equality ∥ v ∥2 = ∥ PU(v)∥2 + ∥ P(v) ∥2 Note: Defining Graphic Design B by means of an orthonormal basis of U ⊥ to an orthonormal basis of Rn. (C) It is to be Q = (u1 ... ur) ∈ M n, R (R) is the matrix of the base vectors u1, ..., ur ∈ B of U as columns. Show that the matrices A and B of PU of PU ⊥ with respect to the standard basis of Rn are given by. A = QQt or B= E-QQt (D) Now let U = 0 ⊂ R3. Calculate the matrix of PU ⊥ with respect to -1 the standard basis of R3 in two different ways: Use one hand the formula for B in (c). Use the other hand, the formula for A from (c), replacing U by U ⊥ and an orthonormal basis for U ⊥ elections.
Consider $\Bbb R^n$ equipped with the standard dot product and let $U \subseteq \Bbb R^n$ be a $r$-dimensional ($0\le r \le n$) linear subspace with orthonormal basis $\mathcal B = \{u_1, \ldots, u_r\}$. Let $P_U: \Bbb R^n \rightarrow \Bbb R^n$ be the associated projection on $U$, given by $$P_U(v) = \sum_{i=j}^r <u_j,v>u_j$$
a) Show that $P_U$ is self adjoint.
b) Let $U^\perp$ be the orthogonal complement of $U$ (see p.9 ex.2) and verify the Pythagorean Theorem for all $v \in \Bbb R^n$, i.e. $$||v||^2 = ||P_U(v)||^2 + ||P_{U^\perp}(v)||^2$$
Hint: Extend $\mathcal B$ via an orthonormal basis of $U^\perp$ to an orthonormal basis of $\Bbb R^n$.
c) Let $Q \in M_{n,r}(\Bbb R)$ be the matrix whose $i$-th column is given by $u_i \in \mathcal B$ for $i = 1,2, \ldots, r$. Show that the matrices $A$ of $P_U$ and $B$ of $P_{U^\perp}$ with respect to the standard basis of $\Bbb R^n$ are given by $$A = QQ^t \ \ \ \ \ \ \ \ \ \ \ B = E-QQ^t$$
d) Now, let $U = \left< \left( \begin{matrix} 1 \\ 0 \\ -1 \end{matrix} \right) \right> \subseteq \Bbb R^3$. Caculate the matrix of $P_{U^\perp}$ with respect to the standard basis of $\Bbb R^3$ in two different ways: First, use the formula in c) to calculate $B$ and then use the formula in c) to calculate $A$, where we replace $U$ with $U^\perp$ and consider an orthonormal basis of $U^\perp$.