Finding Scalar Multipliers for Vector Equation

[fsm]

Homework Statement

Find a and b such that v=au + bw, where u=<1, 2> and w=<1, -1>

Homework Equations

v=au + bw
v=<2, 1>

The Attempt at a Solution

No attempt

I really don't know where to begin. There is not an example like this in the book.

 

[cristo]

Well, try writing the equations in this format:
$$\left(\begin{array}{c}2\\1\end{array}\right)=a\left(\begin{array}{c}1\\2\end{array}\right)+b\left(\begin{array}{c}1\\-1\end{array}\right)$$

Then set up equations for the x (top) component, and the y (bottom) component, and solve for a and b.

 

[HallsofIvy]

Your equation says that a<1, 2>+ b< 1, -1>= <2, 1> or
<a+ b, 2a- b>= <2, 1>. Since two vectors are equal only if corresponding components are equal, you have the two equations a+ b= 2, 2a- b= 1. Solve those for a and b.

 

[fsm]

I actually had it set up like that but thought I was doing something wrong. I didn't know what to do next. Thanks for the help!