Question

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

Short answer

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

Step-by-step solution

To mention given data

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

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

To find the dimension of the space

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 ${ℝ}^n$.

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