**Determine whether the set {[1,2,-1,6], [3,8,9,10],[2,-1,2,-2]} is linearly independent.**

**3. The attempt at a solution**

I construct

[tex]A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 3 & 8 & 9 & 10 \\ 2 & -1 & 2 & -2 \end{array}\right][/tex]

The row echelon form is

[tex]A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 0 & 1 & 6 & -4 \\ 0 & 0 & 1 & -1 \end{array}\right][/tex]

Now there is a theorem saying that if the "rank" of V is smaller than the number of vectors in the set under consideration (i.e., number of rows of V) then the vectors are linearly dependent; otherwise they're independent.

I can't understand this step, how do we determine the "rank" of V?

Furthermore, I have another question;

There is a property that states: "if the set contains more vectors than the dimension of its member vectors, the vectors are linearly dependent." They are thus NOT linearly independent.

So, what if the set contains fewer vectors than the dimension of its member vectors?? Here in my problem I have 3 vectors which are of the 4th dimensions, what does that tell us?

Thanks!

I construct

[tex]A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 3 & 8 & 9 & 10 \\ 2 & -1 & 2 & -2 \end{array}\right][/tex]

The row echelon form is

[tex]A = \left[\begin{array}{ccccc} 1 & 2 & -1 & 6 \\ 0 & 1 & 6 & -4 \\ 0 & 0 & 1 & -1 \end{array}\right][/tex]

Now there is a theorem saying that if the "rank" of V is smaller than the number of vectors in the set under consideration (i.e., number of rows of V) then the vectors are linearly dependent; otherwise they're independent.

I can't understand this step, how do we determine the "rank" of V?

Furthermore, I have another question;

There is a property that states: "if the set contains more vectors than the dimension of its member vectors, the vectors are linearly dependent." They are thus NOT linearly independent.

So, what if the set contains fewer vectors than the dimension of its member vectors?? Here in my problem I have 3 vectors which are of the 4th dimensions, what does that tell us?

Thanks!