If possible, express b = [−7, 1, 5] as a linear combination of a1 =[ 2, −1, 4] , a2 = [3, 0, 1] , and a3 =[ 1, 2, −1] . (if b cannot be written as a linear combination of the other three vectors, enter dne all three answer blanks.)

  b = 3(a1) -5(a2) +2(a3)

Step-by-step explanation:

There are a couple of ways to do this. You can write the system of equations that needs to be solved to find the coefficients of a1, a2, and a3. Let those be x1, x2, and x3. Then you need ...

  2x1 +3x2 +1x3 = -7

  -1x1 +0x2 +2x3 = 1

  4x1 +1x2 -1x3 = 5

Solving this system in any of the usual ways gives ...

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Another way to do the same thing is to compute the inverse of the matrix of vectors a1, a2, a3 and post-multiply the vector b by that. (Each of these vectors is a row of the matrix.)

Either way, the linear combination you want is ...

  b = 3a1 -5a2 +2a3