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Found in: Page 144

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# A subspace ${\mathbf{V}}$ of ${{\mathbf{ℝ}}}^{{\mathbf{n}}}$ is called a hyperplane if ${\mathbit{V}}$ 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 ${{\mathbit{c}}}_{{\mathbf{i}}}$ is nonzero. What is a dimension of a hyperplane in ${{\mathbf{ℝ}}}^{n}$ ? Justify your answer carefully. What is a hyperplane in ${{\mathbf{ℝ}}}^{{\mathbf{3}}}$ ? What is it in ${{\mathbf{ℝ}}}^{{\mathbf{2}}}$?

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

A hyperplane in ${ℝ}^{2}$ is a line.

A hyperplane in ${ℝ}^{3}$ is a plane.

See the step by step solution

## Step 1: To mention the given data and recall the theorem 3.3.7

We have the subspace $V$ of ${\mathrm{ℝ}}^{\mathrm{n}}$ 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 $1×n$ matrix $A=\left[\begin{array}{cccc}{c}_{1}& {c}_{2}& \cdots & {c}_{n}\end{array}\right]$.

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

Theorem 3.3.7, is stated as follows:

For any$n×m$matrix $A$, the equation,

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

## Step 2: To find the dimension of a hyperplane

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

A hyperplane in ${ℝ}^{2}$ is a line.

A hyperplane in ${ℝ}^{3}$ is a plane.