(Solution) - Let B e1 e2 en be an orthonormal basis -(2025 Original AI-Free Solution)

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Let B = {e1, e2,..., en} be an orthonormal basis of an inner product space V. Given T: V ? V, define T': V ? V by

(a) Show that (aT)' = aT'.
(b) Show that (5 + T)' = S' + T'
(c) Show that MB(T') is the transpose of MB(T).
(d) Show that (T')' = T, using part (c).
[Hint: Mb(S) = MB(T) implies that S = T]
(e) Show that (ST)' = T'S', using part (c).
(f) Show that T is symmetric if and only if T = T' [Hint: Use the expansion theorem and Theorem 3.]
(g) Show that T+ T' and TT' are symmetric, using parts (b) through (e).
(h) Show that T'(v) is independent of the choice of orthonormal basis B. [Hint: If D = {f1,..., fn} is also orthonormal, use the fact that e, =