Question

Write x as a linear combination of a and b

Answer 1

v = Ax + By

[6,2] = A[-4,-1] + B[-5, -1]

 

where A and B are the constants you're trying to find.

 

You can split this into two equations:

6 = -4A - 5B

2 = -1A - 1B

 

You can solve this with either substitution or elimination. I'll use elimination by multiplying the second equation by -4 and adding.

 

6 = -4A - 5B

-8 = 4A + 4B

-2 = -B

B = 2

 

2 = -A -(2)

4 = -A

A = -4

 

This means you can write v as

 

v = -4x + 2y

 

And you can check by plugging the vectors in.

[6,2] ?= -4*[-4,-1] + 2*[-5, -1]

[6,2] ?= [16, 4] + [-10, -2]

[6,2] =[6, 2] √