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Q35E

Expert-verified

Linear Algebra With Applications

Found in: Page 144

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: To mention given data

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

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

Step 2: To find the dimension of the space

Now, the vector $\vec{x} ⊥ \vec{υ}$ .

Therefore,

$\begin{array}{l} \vec{υ} \cdot \vec{x} = 0 \\ \Rightarrow υ_{1} x_{1} + υ_{2} x_{2} + \dots + υ_{n} x_{n} = 0 \end{array}$ ,

where, $υ_{i}$ are the components of the vector $\vec{υ}$ .

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 $n - 1$ .

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