1.4.20 Matrix transformations

In the space ( mathbb{V}_{3} ) of geometric vectors with the usual scalar product, the basis vectors ( S_{1}=left{e_{1}, e_{2}, e_{3} ight} ) are specified by coordinates in the basis ( i, j, k ). 1) Find the Gram matrix ( G_{1} ) of the scalar product in this basis. Write down a formula for the length of a vector in terms of its coordinates in the basis ( S_{1} ). 2) Orthogonalize the basis ( S_{1} ). Check the orthonormality of the constructed basis ( S_{2} ) in two ways: a) writing out the coordinates of the vector from ( S_{2} ) in the canonical basis ( i, j, k ). b) making sure that the transformation of the Gram matrix during the transition from the basis ( S_{1} ) to the basis ( S_{2} ) (according to the formula ( G_{2}=P^{T} G_{1} P ), where ( P ) is the transition matrix from the basis ( S_{1} ) to the basis ( S_{2} )) leads to the identity matrix. ( e_{1}=(1,0,2), quad e_{2}=(2,1,1), quad e_{3}=(1,1,0) ).