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 \(B\).