Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

**A subspace $V$ of $ℝ^{n}$ is called a hyperplane if $V$ is defined by a homogeneous linear equation

$c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n} = 0$ ,**

where at least one of the coefficients $c_{i}$ is nonzero. What is a dimension of a hyperplane in $ℝ^{n}$ ? Justify your answer carefully. What is a hyperplane in $ℝ^{3}$ ? What is it in $ℝ^{2}$ ?

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.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

We have the subspace $V$ of $ℝ^{n}$ defined by,

$c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n} = 0$ ,

where at least one of the $c_{i}$ ’s is nonzero.

Therefore, $V$ can be written as

$V = \ker (A)$ , where $A$ is $1 \times n$ matrix $A = [\begin{matrix} c_{1} & c_{2} & \dots & c_{n} \end{matrix}]$ .

Suppose the column vector be $[\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}]$ .

Theorem 3.3.7, is stated as follows:

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

. $\dim (\ker (A)) + \dim (im (A)) = m$

Step 2: To find the dimension of a hyperplane

We have the rank of $A$ $r a n k (A) = 1$ .

Thus, by Theorem 3.3.7, we have,

$\begin{matrix} \dim (V) = \dim (\ker (A)) \\ = n - r a n k (A) \\ = n - 1 \end{matrix}$

$∴ \dim (V) = n - 1$

Therefore,

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

A hyperplane in $ℝ^{2}$ is a line.

A hyperplane in $ℝ^{3}$ is a plane.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

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

Step 2: To find the dimension of a hyperplane

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Linear Algebra With Applications

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Short Answer

A subspace V of ℝn is called a hyperplane if V is defined by a homogeneous linear equation c1x1+c2x2+…+cnxn=0, where at least one of the coefficients ci is nonzero. What is a dimension of a hyperplane in ℝn ? Justify your answer carefully. What is a hyperplane in ℝ3 ? What is it in ℝ2?

Step by Step Solution

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

Step 2: To find the dimension of a hyperplane

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