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Problem 311

Show geometrically that for each angle \(\theta\), the transformation \(\mathrm{T}_{\theta}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\), defined by $(\mathrm{x}, \mathrm{y}) \mathrm{T}_{\theta}=(\mathrm{x} \cos \theta-\mathrm{y} \sin \theta, \mathrm{x} \sin \theta+\mathrm{y} \cos \theta)$, is an orthogonal transformation.

Expert verified

The transformation Tθ represented by the matrix \( \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \) is an orthogonal transformation. It is verified by showing that the magnitudes of the transformed vectors are equal to those of the original vectors (length preservation), and by confirming that the dot product of transformed vectors is equal to the dot product of original vectors (angle preservation).

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Chapter 14

Let $\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{n}}\right),\left(\mathrm{y}_{1}, \mathrm{y}_{2}, \ldots, \mathrm{y}_{\mathrm{n}}\right),\left(\mathrm{z}_{1}, \mathrm{z}_{2}, \ldots \ldots \mathrm{z}_{\mathrm{n}}\right)$ be three vectors in \(\mathrm{R}^{\mathrm{n}}\). Verify the following properties of the dot product using these vectors : a) \(\mathrm{X} \cdot \mathrm{X} \geq 0 ; \mathrm{X} \cdot \mathrm{X}=0\) if and only if \(\mathrm{X}=0\) b) \(X \cdot Y=Y \cdot X\) c) $(\mathrm{X}+\mathrm{Y}) \cdot \mathrm{Z}=\mathrm{X} \cdot \mathrm{Y}+\mathrm{X} \cdot \mathrm{Z}$ d) $(\mathrm{CX}) \cdot \mathrm{Y}=\mathrm{X} \cdot(\mathrm{cY})=\mathrm{c}(\mathrm{X} \cdot \mathrm{Y})$

Chapter 14

Consider the vector space \(\mathrm{C}[0,1]\) of all continuous functions defined on \([0,1]\). If \(\mathrm{f} \in \mathrm{C}[0,1]\), show that $\left({ }^{1} \int_{0} \mathrm{f}^{2}(\mathrm{x}) \mathrm{d} \mathrm{x}\right)^{1 / 2}$ defines a norm on all elements of this vector space.

Chapter 14

Show that the functions $1, \cos \pi \mathrm{x}, \cos 2 \pi \mathrm{x}, \ldots, \cos \mathrm{n} \pi \mathrm{x}$, form an orthogonal set over \([0,1]\). Then normalize them to obtain an orthogonal set.

Chapter 14

Let \(\mathrm{S}=\operatorname{Sp}[(1,0,1),(0,2,1)]\). Then \(\mathrm{S}\) is a subspace of \(\mathrm{R}^{3}\). Find the orthogonal complement of \(\mathrm{S}\).

Chapter 14

Let \(\mathrm{u}=(2,-1,3)\) and \(\mathrm{v}=(4,-1,2)\) be vectors in \(\mathrm{R}^{3}\). Find the orthogonal projection of \(\mathrm{u}\) on \(\mathrm{v}\) and the component of \(\mathrm{u}\) orthogonal to v.

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