Short Answer

Let A be an $n \times m$ matrix. Is the formula ${(kerA)}^{⊥} = im (A^{T})$ necessarily true? Explain.

Yes.

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: Determine the property of orthogonal.

Consider a $n \times m$ matrix A.

Theorem: For any matrix B, $im {(B)}^{⊥} = {kerB}^{T}$ .

By the definition, $im {(A^{T})}^{⊥} = {kerA}^{T^{T}}$ .

As role="math" localid="1659498837555" $A^{T^{T}} = A$ and $A^{⊥^{⊥}} = A$ , take $⊥$ both side in the equation $i m {(A^{T})}^{⊥} = k e r A^{T^{T}}$ as follows.

$i m {(A^{T})}^{⊥} = k e r A^{T^{T}} i m {(A^{T})}^{⊥} = k e r A {\{i m {(A^{T})}^{⊥}\}}^{⊥} = {(k e r A)}^{⊥} i m (A^{T}) = {(k e r A)}^{⊥}$

Hence, the statement is true

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Linear Algebra With Applications

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Short Answer

Let A be an $n \times m$ matrix. Is the formula ${(kerA)}^{⊥} = im (A^{T})$ necessarily true? Explain.

Step by Step Solution

Step 1: Determine the property of orthogonal.

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Linear Algebra With Applications

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Let A be an n×m matrix. Is the formula (kerA)⊥=im(AT) necessarily true? Explain.

Step by Step Solution

Step 1: Determine the property of orthogonal.

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Let A be an $n \times m$ matrix. Is the formula ${(kerA)}^{⊥} = im (A^{T})$ necessarily true? Explain.