Covectors and Scalar Products

Let $V$ be a vector space over the field $K$, $B:=(b_i)$ a basis sequence of $V$ and $f\in V^*$. There exist field elements $\lambda_i$ such that $f=\lambda_ib^{*i}$ which yields if we calculate $f(b_i)$: $$f(b_i)=(\lambda_jb^{*j})(b_j) = \lambda_j\delta^j_i=\lambda_i$$ Also any vector $v\in V$ can be expressed as a linear combination of coordinates: $v=v^ib_i$ and if we apply the linear form $f$ we get $$f(v)=(\lambda_ib^{*i})(v^jb_j)=\lambda_iv^j\delta^i_j=\lambda_iv^i=<[v]_B,(f(b_i))_{i=1,2,...,\mathrm{dim}V} >$$ where $[v]_B$ is the coordinate vector of $v$ regarding the basis $B$.