Can the vector $$\vec{w}=(-5,2)$$ be expressed as a linear combination of $$\vec{u}=(-1,2)$$ and $$\vec{v}=(1,2)$$?
Development:
We want to find $$\lambda$$ and $$\mu$$ so that $$\vec{w}=\lambda\vec{u}+\mu\vec{v}$$. We have: $$$(-5,2)=\lambda(-1,2)+\mu(1,2)=(-\lambda,2\lambda)+(\mu,2\mu)= (-\lambda+\mu,2\lambda+2\mu)$$$ De manera que: $$$\left. \begin{array}{r} -\lambda+\mu=-5 \\ 2\lambda+2\mu=2 \end{array} \right\} \Rightarrow \lambda=3, \ \mu=-2$$$ We have just found values for $$\lambda$$ and $$\mu$$ for which $$\vec{w}=\lambda\vec{u}+\mu\vec{v}$$. Therefore, we can express $$\vec{w}=(-5,2)$$ as a linear combination of $$\vec{u}=(-1,2)$$ and $$\vec{v}=(1,2)$$.
Solution:
The vector $$\vec{w}=(-5,2)$$ can be espress as a linear combination of $$\vec{u}=(-1,2)$$ and $$\vec{v}=(1,2)$$: $$\ \vec{w}=3\vec{u}-2\vec{v}$$.