linear independence of some vectors

Question

If vectors {a1,a2,a3} are linearly dependent and vectors {a2,a3,a4} are linearly independent. how to prove that a1 is a linear combination of {a2,a3} and a4 is not a linear combination of {a1,a2,a3}?

Answer 1

If $\alpha_{1}a_{1} + \alpha_{2}a_{2} + \alpha_{3}a_{3} = 0$ with some $\alpha_{i} \neq 0$. If $\alpha_{1} \neq 0$, then a1 is a linear combination of $a_{2},a_{3}$. IS IT POSSIBLE THAT $\alpha_{1} \neq 0$ ? IF YES, WHAT IS YOUR ANSWER?

Suppose that $$ a_{4} = c (a_{1} + a_{2} + a_{3}) $$ where $c \neq 0$. How $a_{2},a_{3},a_{4}$ are LI, then $$ 0 = \beta_{1} a_{2} + \beta_{2} a_{3} + \beta_{4} = \beta_{1} a_{2} + \beta_{2} a_{3} + c (a_{1} + a_{2} + a_{3}) $$ with $\beta_{j} ,c\neq 0$ for all $j=1,2,3$. Soon, $a_{1},a_{2},a_{3}$ are LI. Contradiction