Inner product, orthogonality and the spectral theorem

Compute the angle between the vectors $u=(1,1,1,1)$ and $v=(1,0,1,0)$ of ${\mathbb{R}}^4$.

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Determine the values of $k\in \mathbb{R}$ for which the vectors $u=(k,1,k)$ and $v=(k,-5,4)$ are orthogonal.

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Compute the orthogonal projection of $v=(3,4,-1)$ onto $u=(1,2,-1)$ and the component of $v$ orthogonal to $u$.

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Verify that the vectors

$$e_1=\frac{1}{3}(1,2,2),e_2=\frac{1}{3}(2,1,-2),e_3=\frac{1}{3}(2,-2,1)$$

form an orthonormal basis of ${\mathbb{R}}^3$ and compute the coordinates of $v=(3,3,3)$ with respect to this basis.

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Apply the Gram-Schmidt process to the vectors $v_1=(1,0,1)$ and $v_2=(1,2,3)$ and find an orthonormal basis of $⟨v_1,v_2⟩$.

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Apply the Gram-Schmidt process to the vectors $v_1=(1,1,1,1)$, $v_2=(1,1,0,0)$, $v_3=(1,0,0,0)$ of ${\mathbb{R}}^4$, in the given order, and find an orthonormal basis of the subspace they span.

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Let $W=\{(x,y,z)\in {\mathbb{R}}^3:x-2y+z=0\}$. Find an orthonormal basis of $W$ whose first vector is parallel to $(1,1,1)$, with the same orientation.

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Determine the dimension and a basis of the orthogonal complement of

$$W=⟨(1,1,0,0),(0,1,1,1)⟩\subseteq {\mathbb{R}}^4.$$

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Let

$$W=\left\{x\in {\mathbb{R}}^4:\{\begin{array}{l}{x}_1+x_2-x_4=0\\ x_2+x_3=0\\ x_1-x_3-x_4=0\end{array}\right\}.$$

Determine $\dim W$ and a basis of $W^{\perp }$.

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Let $W=\{(x,y,z)\in {\mathbb{R}}^3:x+y-z=0\}$. Compute the orthogonal projection of $v=(2,3,2)$ onto $W$ and the distance of $v$ from $W$.

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Determine $a,b,c\in \mathbb{R}$ so that the matrix

$$A=\frac{1}{3}\left(\begin{array}{ccc}1& 2& a\\ 2& 1& b\\ 2& -2& c\end{array}\right)$$

is orthogonal with $\det A=1$.

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Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P^{T}AP=D$, where

$$A=\left(\begin{array}{cc}7& 6\\ 6& -2\end{array}\right).$$

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Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P^{T}AP=D$, where

$$A=\left(\begin{array}{ccc}1& -2& -4\\ -2& 4& -2\\ -4& -2& 1\end{array}\right).$$

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