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Q33E

Expert-verifiedFound in: Page 144

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**A subspace ${\mathbf{V}}$ of ${{\mathbf{\mathbb{R}}}}^{{\mathbf{n}}}$ is called a hyperplane if ${\mathit{V}}$ is defined by a homogeneous linear equation**

** **

**${{\mathbf{c}}}_{{\mathbf{1}}}{{\mathbf{x}}}_{{\mathbf{1}}}{\mathbf{+}}{{\mathbf{c}}}_{{\mathbf{2}}}{{\mathbf{x}}}_{{\mathbf{2}}}{\mathbf{+}}{\mathbf{\dots}}{\mathbf{+}}{{\mathbf{c}}}_{{\mathbf{n}}}{{\mathbf{x}}}_{{\mathbf{n}}}{\mathbf{=}}{\mathbf{0}}$,**

** **

**where at least one of the coefficients ${{\mathit{c}}}_{{\mathbf{i}}}$ is nonzero. What is a dimension of a hyperplane in ${{\mathbf{\mathbb{R}}}}^{n}$ ? Justify your answer carefully. What is a hyperplane in ${{\mathbf{\mathbb{R}}}}^{{\mathbf{3}}}$ ? What is it in ${{\mathbf{\mathbb{R}}}}^{{\mathbf{2}}}$?**

** **

The dimension of a hyperplane in ${\mathbb{R}}^{n}$ is $n-1$ .

A hyperplane in ${\mathbb{R}}^{2}$ is a line.

A hyperplane in ${\mathbb{R}}^{3}$ is a plane.

We have the subspace $V$ of ${\mathrm{\mathbb{R}}}^{\mathrm{n}}$ defined by,

${\mathrm{c}}_{1}{\mathrm{x}}_{1}+{\mathrm{c}}_{2}{\mathrm{x}}_{2}+\cdots +{\mathrm{c}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}=0$,

where at least one of the ${c}_{i}$’s is nonzero.

Therefore, $V$ can be written as

$\mathrm{V}=\mathrm{ker}\left(\mathrm{A}\right)$, where $A$ is $1\times n$ matrix $A=\left[\begin{array}{cccc}{c}_{1}& {c}_{2}& \cdots & {c}_{n}\end{array}\right]$.

Suppose the column vector be$\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{n}\end{array}\right]$ .

Theorem 3.3.7, is stated as follows:

For any$n\times m$matrix $A$, the equation,

.$\mathrm{dim}\left(\mathrm{ker}\left(\mathrm{A}\right)\right)+\mathrm{dim}\left(\mathrm{im}\left(\mathrm{A}\right)\right)=\mathrm{m}$

We have the rank of $A$ $rank\left(A\right)=1$.

Thus, by Theorem 3.3.7, we have,

$\begin{array}{c}\mathrm{dim}\left(V\right)=\mathrm{dim}\left(\mathrm{ker}\left(A\right)\right)\\ =n-rank\left(A\right)\\ =n-1\end{array}$

$\therefore \mathrm{dim}\left(V\right)=n-1$

Therefore,

A hyperplane in ${\mathrm{\mathbb{R}}}^{\mathrm{n}}$ is a subspace of dimension $\mathrm{n}-1$.

A hyperplane in ${\mathbb{R}}^{2}$ is a line.

A hyperplane in ${\mathbb{R}}^{3}$ is a plane.

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