Given the vectors:
u = [3, 0]
v = [-1, 2]
a) Express the vector w = [2, 8] as a linear combination of vectors u and v.
How do I do this?
let [2,8] = a[3,0] + b[-1,2]
so you get 3a - b = 2
and 2b = 8
from the second equation b = 4, subbing that back in to the first...
3a - 4 = 2
a = 2
check to see if [2,8] = 2[3,0] + 4[-1,2]
sure does!
We were lucky that in the second equation one of the variables dropped out, normally you would end up solving 2 equations in 2 unknowns.
What? I don't understand what you did. . .
To express the vector w = [2, 8] as a linear combination of vectors u and v, we need to find scalar coefficients such that w = a*u + b*v, where a and b are scalars.
We can set up a system of equations using the components of the vectors:
2 = a*3 + b*(-1)
8 = a*0 + b*2
We can rewrite the first equation as:
2 = 3a - b
Now we have a system of linear equations. To solve it, we can use any method such as substitution, elimination, or matrix algebra.
Let's use substitution to solve this system:
From the second equation, we can solve for b:
b = 4
Now substitute this value of b into the first equation:
2 = 3a - 4
Rearrange the equation to solve for a:
3a = 6
a = 2
Therefore, the vector w = [2, 8] can be expressed as a linear combination of vectors u and v as:
w = 2*u + 4*v