Math Problems Vault

Problem 26.55

Let $x^{\prime }$ be the orthogonal projection of the vector $x$ onto the subspace $\mathcal{L}$ . Prove that the angle between the vector $x$ and the subspace $\mathcal{L}$ equals the angle between $x$ and $x^{\prime }$ if $x^{\prime } \neq o$ , and equals $\pi / 2$ if $x^{\prime } = o$ .

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Problem 26.56

Let $x^{\prime }$ be the orthogonal projection of the vector $x$ onto the subspace $\mathcal{L}$ , and let the angle between the vector $x$ and the subspace $\mathcal{L}$ equal $\varphi$ . Prove that $\cos \varphi = \left|x^{\prime }\right| / \left|x\right|$ .

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Problem 26.57

Prove that the sum of the angles of the vector $x$ with the subspaces $\mathcal{L}$ and $\mathcal{L}^{\perp }$ equals $\pi / 2$ .

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Problem 26.58

(1)(2)(3)

In an orthonormal basis the vectors $x$ and $f_{1}, \ldots , f_{k}$ are given by their coordinate columns $\xi$ and $\varphi_{1}, \ldots , \varphi_{k}$ . Find the angle between the vector $x$ and the subspace $\mathcal{L}$ spanned by $x$ and $f_{1}, \ldots , f_{k}$ :

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(1)

$\xi =\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} 5 \\ 0 \\ - 4 \\ 2\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} 3 \\ 1 \\ - 5 \\ 0\end{smallmatrix}\right]$ ;

(2)

$\xi =\left[\begin{smallmatrix} 1 \\ 1 \\ 3 \\ 3\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} -4 \\ 4 \\ 2 \\ 3\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} -5 \\ 2 \\ 1 \\ 0\end{smallmatrix}\right]$ ;

(3)

$\xi =\left[\begin{smallmatrix} 2 \\ 6 \\ 2 \\ 6\end{smallmatrix}\right], \varphi_{1} =\left[\begin{smallmatrix} 0 \\ - 1 \\ - 1 \\ 1\end{smallmatrix}\right], \varphi_{2} =\left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ - 1\end{smallmatrix}\right], \varphi_{3} =\left[\begin{smallmatrix} -3 \\ 1 \\ 1 \\ 0\end{smallmatrix}\right]$ ;