Base change/Exchange lemma/Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space, with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$. Let ${}w\in V$ be a vector with a representation

${}w=\sum _{i=1}^{n}s_{i}v_{i}\,,$

where ${}s_{k}\neq 0$ for a certain ${}k$. Let

${}{\mathfrak {w}}=v_{1},\ldots ,v_{k-1},w,v_{k+1},\ldots ,v_{n}\,.$

Determine the transformation matrices ${}M_{\mathfrak {w}}^{\mathfrak {v}}$ and ${}M_{\mathfrak {v}}^{\mathfrak {w}}$.

Create a solution