Find the scalars such that $v_4$ can be written as $v_4=c_1v_1+c_2v_2+c_3v_3$

Question

Find the scalars such that $v_4$ can be written as $v_4=c_1v_1+c_2v_2+c_3v_3$, where $$v_1=(1, 1, 1),\quad v_2=(1, 2, 3),\quad v_3=(1,1,2),\quad v_4=(2,1, 3).$$

Answer 1

Hint. The equation $v_4=c_1v_1+c_2v_2+c_3v_3$ implies that component-wise the following equations hold: $$c_1+c_2+c_3=2,\quad c_1+2c_2+c_3=1,\quad c_1+3c_2+2c_3=3.$$ Can you take it from here?

Note that by subtracting the first and the second equations from the third one, we find immediately that $c_1=0$.

So it remains to solve: $$c_2+c_3=2,\quad 2c_2+c_3=1.$$

Answer 2

You have to argue componentwise:

By looking at the first component you obtain the equation $$c_1\cdot 1+c_2\cdot 1+c_3\cdot 1=2.$$ From the second $$c_1\cdot 1+c_2\cdot 2+c_3\cdot 1=1$$ and from the third $$c_1\cdot 1+c_2\cdot 3+c_3\cdot 2=3$$ All of these must be satisfied at the same time, so they build a linear system. Solve this by the techniques you know.

Answer 3

Let $e_1=(1,0,0),\space e_2=(0,1,0),\space e_3=(0,0,1)$

By some observations we can find that:- $$e_3=v_3-v_1$$ $$e_2=(v_2-v_1)-2e_3=v_1+v_2-2v_3$$ $$e_1=v_1-e_2-e_3=v_1+v_3-v_2$$

We know $v_3=2e_1+e_2+3e_3$. Substitute above expressions and simplify to calculate the coefficients.