Let u= (2,1,0,1,-1) and v=(-2, 3, 1, 0,2). Find scalars a and b so that au-bv=(-8,8,3,-1,7)
au-bv = (2(a+b),a-3b,...) = (-8,8,...)
so,
2a+2b = -8
a-3b = 8
a=-1
b=-3
That fits the first two values. Does it also match the others?
yes. *whew*
To find scalars a and b such that au-bv=(-8, 8, 3, -1, 7), we can set up a system of equations by equating the corresponding components of both sides of the equation.
Given:
u = (2, 1, 0, 1, -1)
v = (-2, 3, 1, 0, 2)
au - bv = (-8, 8, 3, -1, 7)
Let's assume the scalars a and b as variables and set up the equations for each component:
2a - (-2b) = -8 -> 2a + 2b = -8 -> a + b = -4 (Equation 1)
a - (3b) = 8 -> a - 3b = 8 (Equation 2)
1a - (1b) = 3 -> a - b = 3 (Equation 3)
1a - (0b) = -1 -> a = -1 (Equation 4)
-1a - (2b) = 7 -> -a - 2b = 7 (Equation 5)
Now we can solve this system of equations to find the values of a and b.
From Equation 4, we get:
a = -1
Substituting -1 for a in Equation 3, we get:
(-1) - b = 3
-b = 3 + 1
-b = 4
b = -4
So, the values of a and b are:
a = -1
b = -4