What is an effective method for determining expressions for vectors as linear combinations?Example question: Given vector p=(-11,7), vector q=(-3,1), and vector r=(-1,2), express each vector as a...

What is an effective method for determining expressions for vectors as linear combinations?

Example question:

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.

Asked on by egan4

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lfryerda | High School Teacher | (Level 2) Educator

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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 

p=3q+2r   (*)

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:

`q=1/3p-2/3r`

and 

`r=1/2p-3/2q`

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:

p=aq+br

Now putting in the vectors and looking at each component equation, we get:

(-11,7)=a(-3,1)+b(-1,2)

which gives the x-equation

-11=-3a-b

and the y-equation 

7=a+2b

The solution of the two equations can easily be found to be a=3, b=2.

The vector combination is p=3q+2r.

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