Hence, by Theorem 2. For convenience, we define span(∅) = {0 }. 1. For x = (1, −2), we have   1 −5 φβ (x) = [x]β = and φγ (x) = [x]γ = . 23 and the fact that ⎞ ⎛ 1 1  −1 2 2 1 1 ⎠ = ⎝1 1 −1 − 12 2 to find [T]β  . 6 Inner Product Spaces (a) Prove that {u1 , u2 , .

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. 18. Next suppose that it is true for equations of order n − 1, and consider an nth-order differential equation with auxiliary polynomial p(t). Then L(V, W) is finite-dimensional of dimension mn. .

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Since H is nonzero, we can choose vectors u, v ∈ V such that H(u, v) = 0. . 352) and the exercises of Section 1. Let x, y ∈ V.

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. (a) T (b) T ⎛ A1 2. 1 0. We show that W⊥ is T-invariant. Hint: Treat each type of elementary row operation separately.

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(b) Show that if V is finite-dimensional, then W = N(T). (See Figure 2. . 225 0. If x is an element of the set A, then we write x ∈ A; otherwise, we write x ∈ A. .

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Suppose that T is the projection on W along some subspace W . . Proof. 6. Proof.

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25. (a) Every family of sets contains a maximal element. . For each matrix A, find a basis for each generalized eigenspace of LA consisting of a union of disjoint cycles of generalized eigenvectors. An orthonormal basis of eigenvectors is       1 1 0 1 0 −1 1 −1 0 1 0 1 √ ,√ ,√ ,√ 0 0 1 2 1 0 2 0 1 2 1 2 with corresponding eigenvalues 1, 1, −1, −1. Then F has no maximal element.

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† Prove that if W is a subspace of a vector space V and w1 , w2 , . INDEX OF DEFINITIONS FOR CHAPTER 7 Companion matrix 526 Cycle of generalized eigenvectors 488 Cyclic basis 525 Dot diagram for Jordan canonical form 498 Dot diagram for rational canonical form 535 Elementary divisor of a linear operator 539 Elementary divisor of a matrix 541 End vector of a cycle 488 Generalized eigenspace 484 Generalized eigenvector 484 Generator of a cyclic basis 525 Initial vector of a cycle 488 Jordan block 483 Jordan canonical basis 483 Jordan canonical form of a linear operator 483 Jordan canonical form of a matrix 491 Length of a cycle 488 Minimal polynomial of a linear operator 516 Minimal polynomial of a matrix 517 Multiplicity of click here for info elementary divisor 539 Rational canonical basis of a linear operator 526 Rational canonical form for a linear operator 526 Rational canonical form of a matrix 541 Appendices APPENDIX A SETS A set is a collection of objects, called elements of the set. This is the class of bilinear forms. 2 2i (a) Prove that T1 and T2 are self-adjoint and that T = T1 + i T2 . 1(b); that is, eiθ is the unit vector that makes an angle θ with the positive real axis.

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Consequently, by the corollary to Theorem E. (2) Sec. If the space vehicle moves toward the star at velocity v, Earthlings (who remain “almost” stationary relative to S) compute the time it takes for the vehicle to reach the star as t = b/v. (h) The solution set of any system of m linear equations in n unknowns is a subspace of Fn .

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4 . . EXERCISES 1. trace() : Return the sum along diagonals of the array. 6 Inner Product Spaces First, we need some notation and two simple lemmas.

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official site 4 20. Let z = a + bi, where a, b ∈ R. Developed by JavaTpoint. .

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Let f (x) be a polynomial of positive degree, and let a ∈ F . .

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