Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q35E (page 264)

TRUE OR FALSE
The formula ${(k e r B)}^{鈯�} = i m (B^{T})$ holds for all matricesB.

Short Answer

Expert verified

The given statement is true.

Step by step solution

Theorem

For any matrix A,

$i m (A)^{鈯�} = (k e r A^{T})$

Use the theorem

According to the above theorem by replacing A by in $i m {(A)}^{鈯�} = (\ker A^{T})$ and vice versa.

$i m {(A^{T})}^{鈯�} = (\ker {(A^{T})}^{T}) i m {(A^{T})}^{鈯�} = (\ker A)$

Now, take orthogonal complement on both sides.

${(i m {(A^{T})}^{鈯�})}^{鈯�} = {(\ker A)}^{鈯�} 鈬� i m (A^{T}) = {(\ker A)}^{鈯�}$

Then, for any matrix B, we have,

$i m (B^{T}) = {(\ker B)}^{鈯�}$

Its shows the statement is true.

Hence, the statement is true.

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91影视

TRUE OR FALSE
The formula ${(k e r B)}^{鈯�} = i m (B^{T})$ holds for all matricesB.

Short Answer

Step by step solution

Theorem

Use the theorem

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91影视

TRUE OR FALSEThe formula (kerB)鈯�=im(BT)holds for all matricesB.

Short Answer

Step by step solution

Theorem

Use the theorem

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TRUE OR FALSE
The formula ${(k e r B)}^{鈯�} = i m (B^{T})$ holds for all matricesB.