Disclaimer: My English vocabulary for the few nouns used here are google results, so no idea if they are the correctly translated equivalent.
Currently sitting on following problem:
I have calculated the eigenvalues, the matrix and the linearly independent rows, leaving me with the homogeneous linear system:
$$-3x+3y+6z=0$$
Now, as I have read, to calculate the fundamental system (?) I correctly calculated that I can add 2 variables/substitutes (has 3 variables and is of rank $\le 1$). This means to get an "explicit" result, I should be allowed to just set two of the three variables (don't know why though).
My problem begins here: how do I know which variables to select, and when do I know when to stop?
I can say (for whatever reason)
$y=0, z=1$ (then $x=2$)
or $\;y=1, z=0\;$ (then $x=1$)
or $\;y=1, z=1\;$ (then $x=3$)
The solution I have been presented by my tutor only lists the first two options and the basis of the eigenspace is $\{(1,1,0),(2,0,1)\}$. Why isn't $(3,1,1)$ part of the base solution? Is it because it is a linear combination/sum of the other two?
I would write your equation -3x+ 3y+ 6z= 0 as x= y+ 2z. That is, we can determine x for any values of y and z. The vector space is a two dimensional subspace of R^3. Taking y= 1, z= 0, x= 1 so one vector is <1, 1, 0>. Taking y= 0, z= 1, x= 2 so another vector is <2, 0, 1>. The space has {<1, 1, 0>, <2, 0, 1>} as basis.
Yes, <3, 1, 1>= <1, 1, 0>+ <2, 0, 1> is a linear combination of <1, 1, 0> and <2, 0, 1> so would not be included in a basis with them.
Of course, there are an infinite number of bases for any vector space. You could also take {<1, 1, 0>, <3, 1, 1>} or {<2, 0, 1>, <3, 1, 1> as basis.
(A basis for a vector space of dimension n has three properties: a) The vectors span the space b) The vectors are independent c) There are n vectors in the set
And any two of those imply the third.)