Orthogonal Complement

Prove that the orthogonal complement $L^{\perp }$ of a linear subspace $L$ of a Euclidean (Hermitian) space $V$ is a linear subspace. In the case of an $n$-dimensional Euclidean space $V$, find the dimension of $L^{\perp }$, if the dimension of $L$ equals $k$.

Let $L$ be a subspace of a finite-dimensional Euclidean space $V$. Prove that the entire space $V$ is the direct sum of $L$ and its orthogonal complement $L^{\perp }: V = L \oplus L^{\perp }$.

Prove the following properties of the orthogonal complement $L^{\perp }$ of a linear subspace $L$ in a finite-dimensional Euclidean space $V$:

Find a basis of the orthogonal complement $L^{\perp }$ of the subspace $L$ spanned by the vectors:

Find the equations defining the orthogonal complement $L^{\perp }$ of the subspace $L$, if $L$ is given by a system of linear homogeneous equations:

In the Euclidean space $\mathbb {R}_{2 n}[x]$ of polynomials of degree at most $2 n$ with the inner product given by the integral

find the orthogonal complement $L^{\perp }$ to the subspace $L = \left\langle 1, x^{2}, x^{4}, \ldots , x^{2 n}\right\rangle$ (the subspace of even polynomials).

In the Euclidean space of square matrices with the inner product

find the orthogonal complement to the subspace of:

Find the orthogonal projection $x^{\| }$ and the orthogonal component $x^{\perp }$ of the vector $x = (9,1,3,-1)$ when projecting onto the subspace $L = \langle (3,0,4,1),(1,1,1,-1),(3,-3,5,5)\rangle$.

Find the orthogonal projection $x^{\| }$ and the orthogonal component $x^{\perp }$ of the vector $x = (2,5,-2,3)$ when projecting onto the subspace $L = \langle (-3,1,1,1),(1,0,1,0),(-1,0,1,1)\rangle$.

Find the orthogonal projection $Y$ and the orthogonal component $Z$ of the matrix $X$ in the Euclidean space of real square matrices of order two with the inner product $(A, B) = \operatorname {tr}\left(A^{\top } B\right) = \sum_{i, j = 1}^{2} a_{i j} b_{i j}$ when projecting onto the subspace $L = \left\langle A_{1}, A_{2}\right\rangle$, where