3. Let V be a finite dimensional inner product space, and suppose that T is a...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
4. Let TV - V be a linear operator on a finite dimensional inner product space V and P be the orthogonal projection of V onto the subspace W of V. a) Show that is invariant under T if and only if PTP = TP. b) Show that w and we are both invariant under 7 If and only if PT = TP
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V → V be the linear operator defined by 0 T(1) = ( ; ;) A. (i) Compute To((.) (ii) Determine whether or not there is an orthonormal basis of eigenvectors B for which [T]is diagonal. If such a basis exists, find one.
3. Consider the inner product space V = M2x2(C) with the Frobenius inner product, and let T:V + V be the linear operator defined by T(A) -1 (i) Compute T:((1+i :)) (ii) Determine whether or not there is an orthonormal basis of eigenvectors 8 for which [T], is diagonal. If such a basis exists, find one.
Let V be a finite-dimensional vector space, and let B be a basis of V. Show that there is an inner product on V for which B is orthonormal.
Suppose V is a finite dimensional inner product space, and dim V
= n.
If is an orthogonal subset
of V, prove that
a. W can be extended to an orthogonal basis for V.
b. is an orthogonal basis
for
c.
2. Suppose that V is an inner product space. (i) Prove that, for any vectors 01, 02 € V, || 0111? + || 0,2||2 = || v1 + v2||2 + || 01 – v2||2 2 (ii) Prove that, for any vectors V1, V2 € V, if v, and v, are orthogonal then || 01 || + || 112 || 2 = || 01 + 02||2.
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
(5) Prove or give a countcrcxample: If A, B E Cnx"are sclf-adjoint, then AB is also self-adjoint. (6) Let V be a finitc-dimensional inner product space over C, and suppose that T E C(V) has the property that T*--T (such a map is called a skew Hermitian operator (a) Show that the operator iT E (V) is self-adjoint (i.c. Hermitian) (b) Prove that T has purely imaginary eigenvalues (i.e. λ ίμ for μ E R). (c) Prove that T has...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...