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QuestionMathematics

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

Full Solution Locked

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]

Step 3:
: Read off the values of α, β, γ, and δ.

From the last augmented matrix, we can see that: \alpha = 1.866666666666667, \beta = - 6.115107913669065, \gamma = 1.380952380952381, \delta = - 9.738095238095238

Final Answer

[\alpha, \beta, \gamma, \delta] = [1.866666666666667, - 6.115107913669065, 1.380952380952381, - 9.738095238095238]

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QuestionMathematics

Answer

Full Solution Locked

Step 1:
: Set up the augmented matrix for the given vectors.

Step 2:
: Perform row operations to find the scalars α, β, γ, and δ.

Step 3:
: Read off the values of α, β, γ, and δ.

Final Answer

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AnswerHelpful

Full Solution Locked

Step 1:: Set up the augmented matrix for the given vectors.

Step 2:: Perform row operations to find the scalars α, β, γ, and δ.

Step 3:: Read off the values of α, β, γ, and δ.

Final Answer

Need Help with Homework?

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QuestionMathematics

Answer

Step 1:
: Set up the augmented matrix for the given vectors.

Step 2:
: Perform row operations to find the scalars α, β, γ, and δ.

Step 3:
: Read off the values of α, β, γ, and δ.