The vi are the components of v in the basis e;. These components are the projections of the vector on the basis vectors Taking the dot product with basis vector e i Uie,ei=U;S Transformation of basis Given two bases e;, ek and a vector v whose components in each of these bases are vi and ik, respectively, we seek to express the components in basis in terms of the components in the other basis. Since the vector is unique Taking the dot product with e But im(em ei)=immi=i; from which we obtain U,(ej.ei) Note that(ej. ei)are the direction cosines of the basis vectors of one basis on the other basis e,e;=lej‖| eill cose eThe vi are the components of v in the basis ei. These components are the projections of the vector on the basis vectors: v = vjej Taking the dot product with basis vector ei: v.ei = vj (ej .ei) = vjδji = vi Transformation of basis Given two bases ei, ˜ek and a vector v whose components in each of these bases are vi and v˜k, respectively, we seek to express the components in basis in terms of the components in the other basis. Since the vector is unique: v = v˜m˜em = vnen Taking the dot product with ˜ei: v.˜ei = v˜m(˜em.˜ei) = vn(en.˜ei) But v˜m(˜ ei) = v˜mδmi = v˜i em.˜ from which we obtain: v˜i = v.˜ei = vj (ej.˜ei) Note that (ej.˜ei) are the direction cosines of the basis vectors of one basis on the other basis: ej .˜ei = �ej��˜ei� cos e �j˜ei 4