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October 15th, 2018, 12:14 AM   #1
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Need some help with this vector equation

Write the vector v1= (3, -2) as a linear combination of the vectors v2 = (1, 3) and V3 = (-1, 0).

All of the (V)'s have this at the top of it ->.

The answers they got were v1 = -2/3v2 - 11/3v3.
I am trying to wonder how they came up with that conclusion. Someone please explain it to me. Thank you!

Last edited by skipjack; October 15th, 2018 at 01:35 AM.
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October 15th, 2018, 12:49 AM   #2
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By inspection, 3v1 + 2v2 = (3*3 + 2, -3*2 + 2*3) = (11, 0) = -11v3.

Hence v1 = (-2v2 - 11v3)/3 = -2/3v2 - 11/3v3.
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October 15th, 2018, 01:16 AM   #3
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Hi, thanks skipjack, but can you explain what's the process or the formula for solving this type of equation? Because I am confused as to how you got the 3v1 + 2v2.

Last edited by skipjack; October 15th, 2018 at 01:34 AM.
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October 15th, 2018, 01:34 AM   #4
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It was easy to see that multiplying v1 by 3 and v2 by 2 would lead to a multiple of v3.

Alternatively, solve a(1, 3) + b(-1, 0) = (3, -2), i.e. a - b = 3 and 3a + 0b = -2.
The second equation implies a = -2/3. The first equation implies b = a - 3 = -11/3.

Any other method of solving simultaneous linear equations could be used.
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