By Ben Smith
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Additional resources for Algebra 2 [Lecture notes]
31. Suppose that U is a finite dimensional Euclidean vector space, and T : U → U is a selfadjoint operator. Then ker T = R(T )⊥ , R(T ) = (ker T )⊥ , U = ker T ⊕ R(T ). 32. (a) Suppose that U , V , W are finite dimensional Euclidean F-spaces. If T :U →V, S :V →W are linear operators, then (ST )∗ = T ∗ S ∗ . (b) Suppose that T : U → V is a linear operator between two finite dimensional Euclidean F-spaces. Then T is invertible if and only if the adjoint T ∗ : V → U is invertible. Moreover, if T is invertible, then (T −1 )∗ = (T ∗ )−1 .
En is an orthonormal basis of the Euclidean space U . Then the coordinates of a vector u ∈ U in this basis are easily computed. More precisely, if u = x1 e1 + · · · + xn en , x1 , . . 5) implies that ∀j = 1, . . , n. 6) = xj ej , ej = xj , where at the last step we used the orthonormality condition which translates to ei , ej = 1, 0, i=j i = j. Applying Pythagora’s theorem we deduce x1 e1 + · · · + xn en 2 = x21 + · · · + x2n = | u, e1 |2 + · · · + | u, en |2 . 14. 12. If u= 2 1 , then the coordinates x1 , x2 of u in this basis are given by 6 4 x1 = u, e1 = + = 2, 5 5 8 3 x2 = u, e2 = − = 1, 5 5 so that u = 2e1 + e2 .
T The sum of the heights of these towers is (k1 + 1) + (k2 + 1) + · · · + (kr + 1) + 1 + · · · + 1 s = (k1 + · · · + kr ) + (r + s) = dim U . =dim R(N ) =dim ker N 36 LIVIU I. 21 implies that they are mutually disjoint and their union is a linearly independent collection of vectors. The above computation shows that the number of elements in the union of these towers is equal to the dimension of U . Thus, this union is a basis of U . 23. A Jordan basis of a nilpotent operator N : U → U is a basis of U consisting of a disjoint union of towers of N .
Algebra 2 [Lecture notes] by Ben Smith