Linear Independence or Dependence of vectors? {(1,1,3);(1,1,2)}

[chumbo]

Hello, i have a test tomorrow and i need to know the answer to this asap.

Are these vectors linear dependent or independent:
1) {(1,1,3);(1,1,2)} What i have so far in this one:

https://gyazo.com/c8b9157bcb74572efca5897536ba89be

2) {(1,1);(1,1);(3,2)}

thanks in advance.

 

[Ishuda]

Hello, i have a test tomorrow and i need to know the answer to this asap.

Are these vectors linear dependent or independent:
1) {(1,1,3);(1,1,2)} What i have so far in this one: https://gyazo.com/c8b9157bcb74572efca5897536ba89be


2) {(1,1);(1,1);(3,2)}

thanks in advance.

For (1), look at your first equation: You have
(a) \(\displaystyle \alpha\, +\, \beta\, =\, 0\)
(b) \(\displaystyle \alpha\, +\, \beta\, =\, 0\)
(c) \(\displaystyle 3\alpha\, +\, 2\beta\, =\, 0\)
(a) and (b) are the same and give
\(\displaystyle \beta\, =\, -\alpha\)
and (c) then becomes
(c) \(\displaystyle 3\alpha\, -\, 2\alpha\, =\, \alpha\, =\, 0\)
Thus if
\(\displaystyle \alpha\, (1,1,3)\, +\, \beta\, (1,1,2)\, =\, 0\)
then \(\displaystyle \alpha\, =\, \beta\, =\, 0\).
What does that say about linear independence?

Now do the same sort of thing for (2). Actually, you should be able to come up with an \(\displaystyle \alpha,\, \beta,\, and \gamma\) by inspection.