Find all values for $\alpha$ for which ($0,2,3$) can be expressed as a linear combination of the following vectors:

Question

($0,2,3$) As a linear combination of ($2,8,0$), ($5,10 \alpha,1$), ($1,1,-1$)

I have Come up with the following equations: $$2x + 5y + z = 0\\8x+10\alpha y+z=2\\y-z=3$$

I narrowed this down to give $z = \frac{62-30\alpha}{10\alpha-23}$

Would I be correct to assume that the only case where it cannot be expressed as a combination of the other vectors is when z is undefined? So when $\alpha=2.3$?

Answer 1

Giving the system we can check directly, we obtain

• $2x + 5y + z = 0\implies 8x + 20y + 4z = 0$
• $8x+23 y+z=2$
• $y-z=3$

from the first and the second

• $3y-3z=2 \implies y-z=\frac23$

which is in contrast with the last one.