What is an effective method for determining expressions for vectors as linear combinations?
Given vector p=(-11,7), vector q=(-3,1), and vector r=(-1,2), express each vector as a linear combination of the other two.
To find a vector as a linear combination of two other vectors, either a guess-and-check method can be used if the combination is relatively obvious, or solving a system of equations to solve for the combination.
Given the vectors p=(-11,7), q=(-3,1) and r=(-1,2), we can use a guess-and-check to see that
This is checked by noting that
(-11,7)=3(-3,1) + 2(-1,2)=(-9,3)+(-2,4)
Once we have this expression (*) for the vectors, we see that:
However, if we could not originally find the correct guess for the first expression, then we need to solve a system of equations. Using the three vectors, we are looking for a linear combination, which means that we are solving for a and b in the vector equation:
Now putting in the vectors and looking at each component equation, we get:
which gives the x-equation
and the y-equation
The solution of the two equations can easily be found to be a=3, b=2.
The vector combination is p=3q+2r.