Question

A subspace $\mathbf{V}$ of ${\mathbf{ℝ}}^{\mathbf{n}}$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${\mathbf{ℝ}}^n$ ? Justify your answer carefully. What is a hyperplane in${\mathbf{ℝ}}^{\mathbf{3}}$ ? What is it in ${\mathbf{ℝ}}^{\mathbf{2}}$?

Short answer

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.

Step-by-step solution

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

$\mathrm{V}=\ker \left(\mathrm{A}\right)$, 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$\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$ .

Theorem 3.3.7, is stated as follows:

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

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

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,

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

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

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.