QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
Suppose that $S=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}\right\}$ where
\mathbf{u}_{1}=\left(\begin{array}{c}
227 \\
- 245 \\
12 \\
111
\end{array}\right), \mathbf{u}_{2}=\left(\begin{array}{c}
248 \\
146 \\
- 46 \\
- 123
\end{array}\right), \mathbf{u}_{3}=\left(\begin{array}{c}
193 \\
276 \\
4 \\
183
\end{array}\right), \mathbf{u}_{4}=\left(\begin{array}{c}
42 \\
- 71 \\
35 \\
172
\end{array}\right) \text { and } \mathbf{v}=\left(\begin{array}{c}
- 38505 \\
- 9668 \\
2067 \\
- 8262
\end{array}\right)
To avoid typing errors, you can copy and past the following sequences to your Maple worksheet to form entries of the vectors or an augmented matrix.
```
227, - 245, 12, 111
248, 146, - 46, - 123
193, 276, 4, 183
42, - 71, 35, 172
- 38505, - 9668, 2067, - 8262
```
The vector $\mathbf{v}$ is in the span of $S$ written in the form
\alpha \mathbf{u}_{1}+\beta \mathbf{u}_{2}+\gamma \mathbf{u}_{3}+\delta \mathbf{u}_{4}
Find a possible set of values for $\alpha, \beta, \gamma, \delta$. Enter the values of $\alpha, \beta, \gamma, \delta$ as a sequence in the box below
[\alpha, \beta, \gamma, \delta]=
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Answer
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Step 1: Set up the augmented matrix for the given vectors.
We want to find a set of scalars, α, β, γ, and δ, such that v is a linear combination of the vectors in S. In other words, we want to solve the following equation: \alpha \mathbf{u}_{1} + \beta \mathbf{u}_{2} + \gamma \mathbf{u}_{3} + \delta \mathbf{u}_{4} = \mathbf{v} To do this, we can set up an augmented matrix with the vectors u\_i as rows and the vector v as the right side of the equation: \left[\begin{array}{cccc|r} 227 & 248 & 193 & 42 & - 38505 \ - 245 & 146 & 276 & - 71 & - 9668 \ 12 & - 46 & 4 & 35 & 2067 \ 111 & - 123 & 183 & 172 & - 8262 \end{array}\right]
Step 2: Perform row operations to find the scalars α, β, γ, and δ.
First, swap rows 1 and 3 to get a leading 1 in the first column: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ - 245 & 146 & 276 & - 71 & - 9668 \ 227 & 248 & 193 & 42 & - 38505 \ 111 & - 123 & 183 & 172 & - 8262 \end{array}\right] Next, perform the following row operations to eliminate the other entries in the first column: - Subtract 227 / 12 times row 1 from row 2: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 272 & - 1043 & - 15953 \ 227 & 248 & 193 & 42 & - 38505 \ 111 & - 123 & 183 & 172 & - 8262 \end{array}\right] - Subtract 111 / 12 times row 1 from row 4: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 272 & - 1043 & - 15953 \ 227 & 248 & 193 & 42 & - 38505 \ 0 & - 189 & 131 & 107 & - 11447 \end{array}\right] Now, eliminate the entries in the second column above the leading 1: - Subtract 248 / 1003 times row 2 from row 3: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 272 & - 1043 & - 15953 \ 0 & 0 & - 141 & 10 & - 1955 \ 0 & - 189 & 131 & 107 & - 11447 \end{array}\right] - Add 189 / 1003 times row 2 to row 4: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 272 & - 1043 & - 15953 \ 0 & 0 & - 141 & 10 & - 1955 \ 0 & 0 & 0 & 107 & - 1047 \end{array}\right] Next, eliminate the entries in the third column above the leading 1: - Divide row 3 by - 141: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 272 & - 1043 & - 15953 \ 0 & 0 & 0 & 107 & - 1047 \end{array}\right] \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 35 & 2067 \ 0 & 1003 & 0 & - 103 & - 1648 \ 0 & 0 & 0 & 107 & - 1047 \end{array}\right] - Subtract 35 times row 4 from row 1: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 0 & 1020 \ 0 & 1003 & 0 & - 103 & - 1648 \ 0 & 0 & 0 & 107 & - 1047 \end{array}\right] Now, eliminate the entries in the fourth column above the leading 1: - Divide row 4 by 107: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 0 & 1020 \ 0 & 1003 & 0 & - 103 & - 1648 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] - Add 103 times row 4 to row 2: \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 0 & 1020 \ 0 & 1003 & 0 & 0 & - 613.8095238095238 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] \left[\begin{array}{cccc|r} 12 & - 46 & 4 & 0 & 1020 \ 0 & 1003 & 0 & 0 & - 613.8095238095238 \ 0 & 0 & 1 & 0 & 1.380952380952381 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] - Subtract 4 times row 3 from row 1 and add 0 times row 3 to row 2: \left[\begin{array}{cccc|r} 12 & - 46 & 0 & 0 & 1012 \ 0 & 1003 & 0 & 0 & - 613.8095238095238 \ 0 & 0 & 1 & 0 & 1.380952380952381 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] - Divide row 1 by 12: \left[\begin{array}{cccc|r} 0 & 0 & 1 & 0 & 1.380952380952381 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] \left[\begin{array}{cccc|r} 0 & 1 & 0 & 0 & - 6.115107913669065 \ 0 & 0 & 1 & 0 & 1.380952380952381 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right] \left[\begin{array}{cccc|r} 1 & 0 & 0 & 0 & 1.866666666666667 \ 0 & 1 & 0 & 0 & - 6.115107913669065 \ 0 & 0 & 1 & 0 & 1.380952380952381 \ 0 & 0 & 0 & 1 & - 9.738095238095238 \end{array}\right]
Final Answer
[\alpha, \beta, \gamma, \delta] = [1.866666666666667, - 6.115107913669065, 1.380952380952381, - 9.738095238095238]
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