Question

Show(prove) that {1,√2,√3} is linearly independent over Q.

Answer 1

\[\text{ That means we want to show that } (a_1,a_2,a_3)\]
\[\text{ can only be (0,0,0) in } \]
\[a_1\cdot 1+a_2\cdot \sqrt{2}+a_3\cdot \sqrt{3}=0\]

Answer 2

ok i know what dependence mean
but i want a solution

Answer 3

This looks like a proof by contradiction.
Like you will write \[b_1=a_2 \sqrt{2}+a_3 \sqrt{3}\]
Assume a2 and a3 are rational. show by contradiction that b1 is not rational.

Answer 4

i just replaces -a1 with b1

Answer 5

where those rationals aren't equal to zero of course.

Answer 6

but you don't get the result that coefficient are zero !

Answer 7

right

Answer 8

we are trying to show that (0,0,0)=(a1,a2,a3) is the solution solution over the rationals
If we assume a2 and a3 are rational and we show that b1 is not rational then we have shown that the only rational result is (0,0,0)