1.4.20 Matrix Transformations

In space (Mathbb {v} _ {3}) of geometric vectors with the usual scalar product, the vectors of the basis (s_ {1} = left {e_ {1}, e_ {2}, e_ {3} {3} ight}) set by the coordinates in the basis (i, j, k). 1) Find a gram matrix (g_ {1}) a scalar work in this basis. Write out the formula for the length of the vector through its coordinates in the basis (s_ {1}). 2) orthogonalize the basis (s_ {1}). Make a check of the orthonomination of the built basis (s_ {2}) in two ways: a) Writing 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) the transition from the basis (s_ {1}) to the basis (s_ {2})) leads to a unit matrix. (E_ {1} = (1,0,2), quad e_ {2} = (2.1,1), quad e_ {3} = (1.1,0)).