linear independence on vector spaces

Question

In a vector space $\mathbb{V}$, my notes define linear dependence on a set of vectors $\left\{\left|x_1\right>, \left|x_2\right>, \left|x_3\right>, \left|x_4\right>..............\left|x_n\right>\right\}$, such that I could find at least one $\left|x_i\right>$ wherein the equation$$\forall \left|x_j\right>\in\mathbb{V},\forall\lambda_{j}\in\mathbb{C}\quad\sum_{j=1\\j\neq i}^{n}\lambda_j\left|x_j\right>=\left|x_i\right>\quad\lambda_{j}\neq0$$ holds true. Now, my question is, if I could find just one $\left|x_i\right>, i\in[1,n]\&i\in\mathbb{Z}$ then, shouldn't I also find the same for all other vectors, $\left|x_j\right>$ where $j\neq i$? Then why is the "at least one" clause put into the definition?

Answer 1

$$\forall \left|x_j\right>\in\mathbb{V},\forall\lambda_{j}\in\mathbb{C}\quad\sum_{j=1\\j\neq i}^{n}\lambda_j\left|x_j\right>=\left|x_i\right>\quad\lambda_{j}\neq0$$

The condition as written, with $\forall\lambda_{j}\in\mathbb{C}$, implies that $\left|x_j\right> = 0$ for all $j=1, 2,\cdots,n$. This is because writing it for a set of $\lambda_j$, then for the correspondent set of $2\,\lambda_j$ it follows that $\left|x_i\right> = 0$. Then writing it for the initial set of $\lambda_j$ with all but one having the sign changed, and adding the two equalities, it follows that the corresponding vector $\left|x_j\right> = 0$.

For comparison, the condition modified as below gives a sufficient condition for the set of $\left|x_j\right>$ to be linearly dependent.

$$\text{given}\; \left|x_j\right>\in\mathbb{V}, \;\exists i \;\;\text{for which} \;\;\exists \lambda_{j}\in\mathbb{C} \setminus{\{0\}}\;\;\text{so that}\;\;\sum_{j=1\\j\neq i}^{n}\lambda_j\left|x_j\right>=\left|x_i\right>$$

Written this way, the condition is symmetric in $i$ in the sense that if it holds for one of the $\left|x_i\right>$ then it will hold for any other $\left|x_j\right>$ by simply dividing by $\lambda_j \ne 0$ and rearranging the terms.

The condition, even modified as above, is however not necessary for linearly dependence in the usual sense. For example, any set of vectors containg the zero vector is linearly dependent, but does not necessarily satisfy the respective condition.