Solving Linear Dependency: u,v,w Equation

[Pengwuino]

Im a little confused here and I have a feeling I forgot how to do matrix operations. The problem is to determine whether or not the following equation is dependant or independent.

$$u = (1, - 1,2),v = (3,0,1),w = (1, - 2,2)$$

I thought that I setup the matrix like this and try to look for the echelon form:

$$\begin{array}{*{20}c} 1 & 3 & 1 \\ { - 1} & 0 & { - 2} \\ 2 & 1 & 2 \\ \end{array}$$

I got it down to:

$$\begin{array}{*{20}c} 1 & 0 & 1 \\ 0 & { - 5} & 0 \\ 0 & 3 & { - 1} \\ \end{array}$$

and I got a little confused, I am not sure what to do… or maybe I screwed up earlier?

 

[HallsofIvy]

If you really want to get it into echelon form, just continue! To get a 0 below that -5, multiply the second row by 3/5 and add to the third row.
Although, with that 0 in the second row, third column, it's already obvious, isn't it, that you will get a non-zero third row. That's enough to show these three vectors are not dependent.

 

[Pengwuino]

Ok I'm still a little confused. What does the matrix have to become in order for me to be able to say its linear independant or linear dependant?

I also ran the matrix through mathematica and it was able to reduce it to an identity matrix... did i maybe do the problem wrong?

 

[Hurkyl]

The goal of the approach you are using is to find the dimension of the space your vectors span.

Can you use that number to tell if your vectors are linearly independent or not?

How does this number relate to the rank of the matrix you created?

What about the rank of the matrix produced by fully row-reducing it?

Can you tell what the rank of the fully row-reduced matrix is?

 

[Pengwuino]

What do rank and span mean? I looked in the book and its farther into the book then the problem is.

 

[shmoe]

If we call your matrix A, your vectors are linearly independant if and only if the only column vector X satisfying AX=0 is the zero vector. This is just the definition of linear independance as AX is just a linear combination of your vectors (the columns of A).

If you can reduce A to the identity matrix, what does this say about solutions to the homogeneous system AX=0?

If you are doubting your result, you might try "plotting" your vectors with a few pencils/straws/sticks/whatever. Do you know what linear independance will mean geometrically here?