Let $a$ be a vector, but not the zero-vector. If the inner product $a \cdot b = a \cdot c$. Are $b$ and $c$ linear dependent if the vectors $a, b$ and $c$ are 2-dimensional?
I would know how to show this if I would know the actual vector, but now I have no clue what to do. How do you argue whether $b$ and $c$ are linear dependent or not?
On
$a \cdot b = a \cdot c \implies a \cdot (b-c) =0 \implies (b-c) \perp a$, if $a$ or $(b-c)$ is not zero. So, it is possible that both $b, c \in S$, where $S$ is defined as follows-
$$S = \left\{ x\in \mathbb{R}^n: x^Ta = 0, \text{ where } a\in \mathbb{R}^n\right\}.$$
But it is also possible that $b,c \notin S$ and two linearly independent vectors, though $(b-c) \in S$. This is because $S$ is a subspace of $\mathbb{R}^n (\text{if } a\neq \theta_n)$.
One of the examples of such cases can be found in Servaes's answer.
Take $a=(1,1)$, $b=(1,2)$ and $c=(2,1)$. Then $a\cdot b=a\cdot c=3$, but $b$ and $c$ are not linearly dependent.