What does a weight vector with weight $(\lambda_1,\cdots,\lambda_n)$ actually mean?
Let $V$ be a $gl(n)$-module and I have that $v\in V$ is a weight vector if it is an eigenvector for all elements of the Cartan subalgebra, and so then let the basis of the Cartan subalgebra be $\{a_i^i:i=1,\cdots,n\}$ and we get $a_i^i v = \lambda_i v$ and we all the eigenvalues get put into this weight vector, yielding us the
$$(\lambda_1,\cdots,\lambda_n)$$
Well what actually is this weight? How should I interpret it? What point is there to knowing this weight vectors weight?
Also while I am at it, what is a fundamental weight?