Q38E TRUE OR FALSEThe formula聽im(B)=... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q38E (page 264)

TRUE OR FALSE
The formula $i m (B) = i m (B^{T} B)$ holds for all square matrices B.

Short Answer

Expert verified

The solution is false.

Step by step solution

Given information

Consider the statement as follows:

鈥淭he formula $i m (B) = i m (B^{T} B)$ holds for all matrices B.鈥�

Explanation of the solution

Consider the matrix B as follows:

$B = [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]$

Then $i m (B)$ as follows:

$i m (B) = \{B [\begin{matrix} x \\ y \end{matrix}] : [\begin{matrix} x \\ y \end{matrix}] 鈭� R^{2}\} = \{|\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}| [\begin{matrix} x \\ y \end{matrix}] : [\begin{matrix} x \\ y \end{matrix}] 鈭� R^{2}\} = [[\begin{matrix} x \\ y \end{matrix}] : x 鈭� R^{2}]$

Now, simplify for $B^{T} B$ as follows:

$B^{T} B = {[\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}]}^{T} [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 2 & 0 \\ 0 & 0 \end{matrix}]$

Now, simplify further as follows.

role="math" localid="1659614309579" $i m (B^{T} B) = \{B^{T} B [\begin{matrix} x \\ y \end{matrix}] : [\begin{matrix} x \\ y \end{matrix}] R^{2}\} = \{[\begin{matrix} 2 x \\ 0 \end{matrix}] : x R^{2}\}$

It can be observed that, $[\begin{matrix} 1 \\ 1 \end{matrix}] 鈭� i m (B)$ , but $[\begin{matrix} 1 \\ 1 \end{matrix}] 鈭� i m (B^{T} B)$ . Then,

$i m (B) 鈭� i m (B^{T} B)$

Hence, the given statement 鈥淭he formula $i m (B) = i m (B^{T} B)$ holds for all matrices B鈥� is false.

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91影视

TRUE OR FALSE
The formula $i m (B) = i m (B^{T} B)$ holds for all square matrices B.

Short Answer

Step by step solution

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Explanation of the solution

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91影视

TRUE OR FALSEThe formulaim(B)=im(BTB) holds for all square matrices B.

Short Answer

Step by step solution

Given information

Explanation of the solution

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TRUE OR FALSE
The formula $i m (B) = i m (B^{T} B)$ holds for all square matrices B.