Linear algebra

Question

Let v1=(1 2 5), v2=(2 3 2). Determine m such that (11 4 m) is a linear combination of (v1, v2)

Answer 1

Let's set this up as a system of equations:

 

x*v1 + y*v2 = (11 4 m) can be expanded to:

 

Eq 1: x + 2y = 11

Eq 2: 2x + 3y = 4

Eq 3: 5x + 2y = m

 

Since m only appears in Eq 3, we can solve equations 1 and 2 independently. 

2*Eq 1: 2x + 4y = 22
-1* Eq 2: -2x - 3y = -4

 y = 18 (by summing the equations)

 

x + 2(18) = 11 (plugging y value back into Eq 1)

x = -25

 

Now let's plug x = -27, y = 18 back into Eq 3:

5(-25) + 2(18) = m

m= -89