SOLUTION: The vector v=[4 5] can be written as a linear combination of the vectors v1=[1 3] and v2=[4 -2] by letting c1=? and c2=?. Find c1 and c2 to solve.

Algebra ->  College  -> Linear Algebra -> SOLUTION: The vector v=[4 5] can be written as a linear combination of the vectors v1=[1 3] and v2=[4 -2] by letting c1=? and c2=?. Find c1 and c2 to solve.      Log On


   

Question 135207: The vector v=[4 5] can be written as a linear combination of the vectors v1=[1 3] and v2=[4 -2] by letting c1=? and c2=?. Find c1 and c2 to solve.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

    v = c1v1 + c2v2

[4 5] = c1[1 3] + c2[4 -2]

[4 5] = [c1*1 c1*3] + [c2*4 c2*-2]

[4 5] = [c1 3c1] + [4c2 -2c2]

[4 5] = [c1+4c2  3c1-2c2]

Equating first components give the equation

4 = c1 + 4c2

Equating second components gives the equation

5 = 3c1 - 2c2

So we have this system of equations:

4 =  c1 + 4c2
5 = 3c1 - 2c2

Or

 c1 + 4c2 = 4 
3c1 - 2c2 = 5

Solve that system of equations and get

c1 = 2, c2 = 1/2

Edwin