- #1

Mutaja

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## Homework Statement

Check if the following set of vectors are linearly dependent or independent:

A) V1= [itex]\stackrel{1}{1}[/itex] V2= [itex]\stackrel{1}{3}[/itex]

B) V1= [itex]\stackrel{\stackrel{1}{2}}{3}[/itex] V2= [itex]\stackrel{\stackrel{2}{1}}{3}[/itex]

C) V1= [itex]\stackrel{1}{3}[/itex] V2= [itex]\stackrel{2}{1}[/itex] V3= [itex]\stackrel{-1}{2}[/itex]

## Homework Equations

Gauss-Jordan, equation with two unknowns.

## The Attempt at a Solution

A) x1 * V1 + x2*v2 = 0

if x1 AND x2 = 0, then they're linearly independent. If not, they're linearly dependent.

[itex]\stackrel{1}{1}[/itex] [itex]\stackrel{1}{3}[/itex] * [itex]\stackrel{X1}{X2}[/itex] = 0

Using gauss jordan I get: [itex]\stackrel{1}{1}[/itex] [itex]\stackrel{1}{3}[/itex] [itex]\stackrel{0}{0}[/itex] -> [itex]\stackrel{1}{0}[/itex][itex]\stackrel{1}{2}[/itex][itex]\stackrel{0}{0}[/itex]-> [itex]\stackrel{1}{0}[/itex][itex]\stackrel{1}{1}[/itex][itex]\stackrel{0}{0}[/itex]-> [itex]\stackrel{1}{0}[/itex][itex]\stackrel{0}{1}[/itex][itex]\stackrel{0}{0}[/itex] which gives me x1 = 0, x2 = 0.

X1+ X2 = 0

X1 + 3X2 = 0

-2X2 = 0 -> X2 = 0

X1 = 0.

Both X1 and X2 has to be 0 -> linearly independent.

B) X1*V1 + X2*V2 = 0

Using the same method -> Gauss Jordan. I end up with the same, X1 an X2 has to be 0 -> linearly independent.

Please do let me know if I should write all my work here as well. Leaving it as it is for now as it takes 15 minutes, and I've done the exact same steps as above.

C) Here we can see that V1 - V2 = V3, therefore they are linearly dependent.

Is this sufficient to "check" if they're linearly dependent or independent, or do I have to do more work? If so, do you have any tips on how to proceed with three vectors?

Also, any tips regarding how to write vectors or matrices efficiently with latex or what it's called, please let me know. I'll be more than happy to include more of my work if I can do it somewhat efficiently.

Thanks for any input.