6. Let T: V → V be a linear operator on a finite-dimensional inner product space V.Show that T is the orthogonal projection onto a subspace of V if and only if T isself-adjoint and T² = T. (Hint for : use spectral theorem and the theorem thateigenvalues of self-adjoint operators are all real.)

Let T:V→Vbe a linear operator on a finite dimensional inner product space V. It has to be proven…

Answered: 6. Let T: V→ V be a linear operator on…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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one example of a 4-dimensional vector space that is not R 4
2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.
show that is not vector space
Consider the operator Son the vector space by S(a+bx) = - a+b+ ( a + 2b) x A) Pick a basis B = { b₁,b₂3. Find the minimal polynomials NT, b, (X), Nr, 0₂ (x), and Ns (x) R₁ [x] given B) Show that S is cyclic by finding a vector v such that
Let T: V-->W be a linear transformation between finite-dimensional vector spaces V and W Let B and C be bases for V and W, respectively, and let A= [ T]C<--B. Use the results of this section to give a matrixbased proof of the Rank Theorem
Let M₂x2 be the vector space of all 2 x 2 matrices and define a linear transformation T: M2x2 → M2x2 by a [id] Describe the kernel of T. C T(A) = A + AT, where A = O No answer text provided. odlo 0 -a ° 0 O [1] :d is a real number} [9] : b is a real number} -b O (1 :d is a real number} (d c is a real number} °41 :d is a real number} : a,b,c,d are real numbers} [9] : b is a real number}
Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1. Show that 1 (v, w1) + 2 (v, w2) + 3 (v, w3) + ...+ k (v, wk) cannot be a negative number.
Q1) Let S (x-3, x² + 2x, x² + 1). Is S is span (P2(x)) or not? Q2) Let T= ((2,-2,4), (3,-5,4), (0,1,1)). Show that whether T is linearly independent or not? Q3) Let N = (vv) be a basis for vector space V Show that every vector in V can be written as uniquely linear combination of vectors of V.
P3 = {ao + a¡t + azt² + a3t³ : ao, a1, az E R} be the vector space consisting of all polynomials of degree less than or equal to 3. Observ basis for P3, where B = {1, t, t², t³}. Let T : P3 - P3 be a linear transformation defined by T(ao + ajt + azt² + a3t°) = a1 + 2a2 + 6azt Find the matrix representation [T]B_of T with respect to B.

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6. Let T: V→ V be a linear operator on a finite-dimensional inner product space V. Show that T is the orthogonal projection onto a subspace of V if and only if T is self-adjoint and T² = T. (Hint for: use spectral theorem and the theorem that eigenvalues of self-adjoint operators are all real.)