Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider a subspace $V$ in $ℝ^{m}$ that is defined by homogeneous linear equations

$| \begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 m} x_{m} = 0 \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 m} x_{m} = 0 \\ ⋮ ⋮ ⋮ ⋮ \\ a_{n 1} x_{1} + a_{22} x_{2} + \dots + a_{nm} x_{m} = 0 \end{array} |$ .

What is the relationship between the dimension of $V$ and the quantity $m - n$
? State your answer as an inequality. Explain carefully.

A subspace $V$ of $ℝ^{m}$ has dimension at least $m - n$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

We have the subspace $V$ of $ℝ^{m}$ defined by $n$ homogeneous linear eqautions,

role="math" localid="1660126682574" $|\begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 m} x_{m} = 0 \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 m} x_{m} = 0 \\ ⋮ ⋮ ⋮ ⋮ \\ a_{n 1} x_{1} + a_{n 2} x_{2} + \dots + a_{n m} x_{m} = 0 \end{array}|$ ,

Therefore, $V$ can be written as

$V = \ker (A)$ , where $A$ is an $n \times m$ matrix $A = [a_{i j}]$ .

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

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 relationship between the dimension V and m-n

We have the rank of $A$ $r a n k (A) \leq n$ .

Thus, by Theorem 3.3.7, we have,

$\begin{matrix} \dim (V) = \dim (\ker (A)) \\ = m - rank (A) \\ \geq m - n \end{matrix}$

$∴ \dim (V) \geq m - n$

Hence, a subspace $V$ of $ℝ^{m}$ has dimension at least $m - n$ .

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

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

Step 2: To find the relationship between the dimension V and m-n

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Linear Algebra With Applications

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Short Answer

Consider a subspace V in ℝm that is defined by homogeneous linear equations |a11x1+a12x2+…+a1mxm=0a21x1+a22x2+…+a2mxm=0 ⋮ ⋮ ⋮ ⋮ an1x1+a22x2+…+anmxm=0|. What is the relationship between the dimension of Vand the quantity m-n? State your answer as an inequality. Explain carefully.

Step by Step Solution

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

Step 2: To find the relationship between the dimension V and m-n

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