Use a linear system to write u = (12, 19, 31) as a linear combination of u1 = (1,1,2), u2 = (2,3,5), and uz = (3,5,8). is w = (1,0,1) a linear com- bination of u1, u2, u3?

Let ,

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is also a linear combination of , , .

Step-by-step explanation:

Write a linear system for , with one equation for each component. The augmented matrix for the first linear system will be:

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Transform this matrix to its reduced row-echelon form using Gaussian Elimination. Solve for each variable.

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Therefore,

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Set up a similar augmented matrix for :

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The second part of this question isn't concerned about the exact value of , , or . Therefore, before proceeding with Gaussian Elimination, start by checking the determinant of the coefficient matrix. If this determinant is nonzero, will always be a unique linear combination of , , now matter what value it takes.

In this case (also as seen in the first part of this question), the determinant of the coefficient matrix for , , and is zero. Determining whether the linear combination is possible will require elimination.

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Similar to the first part of this question, this linear system is consistent. . is indeed a linear combination of , , .

add the numbers under frequency to find the total number:

5 + 8 +7 + 4 + 0 +1 = 25

the answer is 25.