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Q38E

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Found in: Page 191

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# (M) Determine whether w is in the column space of $$A$$, the null space of $$A$$, or both, where$${\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)$$

$${\mathop{\rm w}\nolimits}$$ is in Col A, and w is in Nul A.

See the step by step solution

## Step 1: Write an augmented matrix

Consider the augmented matrix $$\left( {\begin{array}{*{20}{c}}A&{\mathop{\rm w}\nolimits} \end{array}} \right)$$ as shown below:

$$\left( {\begin{array}{*{20}{c}}A&w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)$$

## Step 2: Convert the matrix into row-reduced echelon form

Consider the matrix $$A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)$$.

Use the code in MATLAB to obtain the row-reduced echelon form as shown below:

$$\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0\,\,\,1;\, - 5\,\,\,2\,\,\,1\,\,\, - 2\,\,\,2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3\,\,\,1;\,3\,\,\, - 2\,\,\,1\,\,\,\,0\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}$$

$$\left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&{ - 1}&0&{ - 2}\\0&1&{ - 2}&0&{ - 3}\\0&0&1&1&1\\0&0&0&0&0\end{array}} \right)$$

## Step 3: Determine whether w is in the column space of A

A typical vector v in Col A has the property that the equation $$A{\mathop{\rm x}\nolimits} = {\mathop{\rm v}\nolimits}$$ is consistent.

The system of the equation of matrix A is consistent.

Thus, $${\mathop{\rm w}\nolimits}$$ is in Col A.

## Step 4: Determine whether w is in the Null space of A

A typical vector v in Nul A has the property that $$A{\mathop{\rm v}\nolimits} = 0$$.

Use the code in the MATLAB to compute the matrix $${\mathop{\rm Aw}\nolimits}$$ as shown below:

$$\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0;\, - 5\,\,\,2\,\,\,1\,\,\, - 2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3;\,3\,\,\, - 2\,\,\,1\,\,\,\,0} \right)\\ > > {\mathop{\rm w}\nolimits} = \left( {1;\,\,2;\,\,1;\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm A}\nolimits} * w\end{array}$$

$$\begin{array}{c}{\mathop{\rm A}\nolimits} {\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}0\\0\\0\\0\end{array}} \right)\end{array}$$

Thus, w is in Nul A.