representation of a vector in an $n$-dimensional vector space $X$ is unique Show that in an $n-$dimensional vector space $X$, the representation of any $x \in X$ as a linear combination of a given basis $\{e_1,...,e_n \}$ is unique.
My attempt:
Let $\{e_i\}$ be a basis for $X$, and take any $x \in X$. Hence there exists constants $\lambda_i \in K$ such that $x = \sum \lambda_i e_i $. Similarly, can find constants $\mu_i \in K$ such that $x = \sum \mu_i e_i$. Hence
$$ \Theta = x -x = \sum_{i=1}^n(\lambda_i - \mu_i)e_i \implies \mu_i - \lambda_i = 0 \implies \mu_i = \lambda_i \; \; \forall i$$
Hence, the representation of $x$ is unique as required.
Is this a correct approach? I am interpreting the problem correctly? thanks

Yes, it's correct. The fact that a basis is a generating family gives the existence of the representation as a linear combination while freeness guarantees, as you notices, uniqueness of this representation.
