I am not sure how to solve a problem, or im not sure what the problem actually asks from me....
The Problem (translated): For which $\alpha$ ($\alpha $ element R) has the vectorsystem rank=3
Then there is a vectorsystem given
$$\Bigg\lbrace \begin{bmatrix}\alpha\\1\\3 \end{bmatrix}, \begin{bmatrix}1\\3\\-2\alpha \end{bmatrix} , \begin{bmatrix}1\\2\\-3 \end{bmatrix}\Bigg\rbrace$$
My problem i think is that i haven’t really understood what rank is or to specify, what the difference between dimension and rank is. I know that rank of a vectorsystem is the dimension of its linear combination. (I’m actually not quite sure that’s the right word for what i mean. In German its „lineare Hülle“.
I tried solving it by writing the vectorsystem as linear combination equaling the zero vector and then solving for alpha in terms of x1, x2, x3. To see for what alpha linear independent vectors are produced, and therefore they should have dimension 3. But rank 3?
It would be so helpfull if you could give me some hint, on how to think of this problem in terms of rank.
Many thanks
The rank is the dimension of the space spanned by thos vectors. So, yes, the rank will be $3$ if and only if every vector of $\mathbb{R}^3$ can be written as a linear combination of those $3$ vectors. This is also equivalent to the assertion that the matrix whose columns are those vectors has non-zero determinant.