Answer to Question #325924 in Linear Algebra for bucie

Question

How is w=(3,5,1)∈R

3

 a linear combination of u=(0,−2,2)

 and v=(1,3,−1)

 ?

Answer 1

w is a linear combination of u and v if exist such x and y so x*u +y*v = w

That means that $\begin{bmatrix} 0 & 1 \\ -2 & 3 \\ 2 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 3 \\ 5 \\ 1 \end{bmatrix}$

Or $\begin{cases} y = 3 \\ -2 x + 3y = 5 \\ 2x - y = 1 \end{cases}$

From the fist equation find that y = 3, and substitute it in the last equation get 2x -3 = 1, or x = 2. After substituting x=2 and y=3 in the second equation confirmed it is satisfied.

So w = 2*u + 3*v