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Q35E

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Found in: Page 144

### Linear Algebra With Applications

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

# Consider a non-zero vector $\stackrel{\mathbf{\to }}{\mathbf{\upsilon }}$ 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 $\stackrel{\mathbf{\to }}{\mathbf{\upsilon }}$?

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

See the step by step solution

## Step 1: To mention given data

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

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

## Step 2: To find the dimension of the space

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

Therefore,

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

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

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 $n-1$.