Question

Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists an invertible n × n matrix B such that A (d) Prove that the map 《, 》; B| B R2x2 x R2x2- R defined by (A, B》 = tr(ATB) defines an inner product on R2x2. Find (cf: part (a)) an isomorphism T : R2x2 -> R4 such that 〈(A, B)〉-T(A)-T(B) for all A, B E R2x2

Answer 1

"Let n elementof N, and V be an n-dimensional vector space. and, Let LeftAngleBracket middot, middot RightAngleBracket: V times V rightarrow R be an inner product on V (a): Let {V_1^Vector, V_2^Vector, hellip V_n^Vector} be an orthogonal basis of V. Then any V^Vector elementof V can be unequally written as: V^Vector = Sigma^n_i = 1 lambda_i V_i^Vector Let {e_1^Vector, e_2^Vector, hellip e_n^Vector} be the standard term of R^n Let us define T: V rightarrow R^n by T (V^Vector) = T(Sigma^n_i = 1 lambda_i V_i^Vector) = (lambda_1 lambda_2 hellip lambda_n) = Sigma^n_i = 1 lambda_i e_i^Vector then for V^Vector = Sigma^n_i = 1 x_i v_i^Vector, w^Vector = Sigma^n_i = 1 mi_i V_i^Vector elementof V and e elementof R we have, T(V^Vector + ew^Vector) = T(Sigma^n_i = 1 lambda_i V_i^Vector + e Sigma^n_i = 1 mu_i V_i^Vector) = T (Sigma^n_i = 1 (lambda_i + e mu_i) V_i^Vector) =