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Q35E

Expert-verifiedFound in: Page 144

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Consider a non-zero vector $\overrightarrow{\mathbf{\upsilon}}$ in $\stackrel{}{{\mathbf{\mathbb{R}}}^{\mathbf{n}}}$. What is the dimension of the space of all vectors in ${{\mathbf{\mathbb{R}}}}^{{\mathbf{n}}}$that are perpendicular to $\overrightarrow{\mathbf{\upsilon}}$?**

The dimension of the space of all vectors in ${\mathbb{R}}^{n}$ is $n-1$.

Let $\overrightarrow{x}\in {\mathbb{R}}^{n}$be a nonzero vector.

We need to find the dimension of the space of all vectors in ${\mathbb{R}}^{n}$ that are perpendicular to $\overrightarrow{\upsilon}$ .

Now, the vector $\overrightarrow{x}\perp \overrightarrow{\upsilon}$.

Therefore,

$\begin{array}{l}\overrightarrow{\mathrm{\upsilon}}\cdot \overrightarrow{\mathrm{x}}=0\\ \Rightarrow {\mathrm{\upsilon}}_{1}{\mathrm{x}}_{1}+{\mathrm{\upsilon}}_{2}{\mathrm{x}}_{2}+\cdots +{\mathrm{\upsilon}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}=0\end{array}$ ,

where, ${\upsilon}_{i}$ are the components of the vector $\overrightarrow{\upsilon}$.

Thus, by Exercise 33, these vectors form a hyperplane in ${\mathbb{R}}^{n}$.

Hence, the dimension of the space of all vectors in ${\mathbb{R}}^{n}$ is $n-1$.

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