Express the vector v as a linear combination of x and y

Question

Express the vector v=[6,2] as a linear combination of x=[-4,-1] and y=[-5,-1].

v=?x+?y

Answer 1

Hi Dalia,

 

To find the linear combination that results in vector v, you can set up your equation and split it into one equation per component of your vectors. This method will work for two of the other questions you've posted, just with 3 and 4 variables in those questions. The algebra for solving those systems of equations is much more complicated, but you should be able to do them as well.

 

In this problem, your vector equation is:

 

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] √