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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 12EQ

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How can I prove that in terms of this inner product the operator ∂t : s(t) → ∂ts(t) is anti-Hermitian.

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Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
The conjugate space H" is also a Hilbert space with respect to the inner product defined by (fx. fy) = (y,x).
Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive (i. e. satisfies f(v+w) = f(v)+f(w) for v,w in C) but not linear as a map of complex vector spaces. Show however that if we define f as the map f : R^2 -> R^2 given by f(x,y) = (x+2y, 3x+4y) then f is linear as a map of real vector spaces.
3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition + : V × V → V defined f: via (f+g)(x) = f(x) + g(x), x = R", and scalar multiplication : Rx V → V defined via (A. f)(x) = \f(x), ● x ER". Now let W = {f: R" → R: f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R" → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.
(2) Is Mmxn(Q) is a vector space over R?
2. Let T be a one-to-one linear transformation from a vector space V into R". For u, v € V, show that the formula = T(u) · T(v) defines an inner product on V.
The nabla formulas corresponding to the derivative rule of income are gVf+fVg, (Vƒ). F + f(V.F), (Vf) x F + f(V x F). ▼(fg) V. (fF) ▼ x (fF) = = = Justify the middle one (e.g. in the case of n = 3). Hint: The coordinate functions of the vector field fF are fF₁ etc
2. Let R3 be a real vector space, and u = (u1, u2, u3), v = = (v1, v2, v3) E R", then show that = U1.V1 + 2 u2.V2 + U3.V3 defines an inner product on R$.
Q.3) Graph the vector field F(x,y) 3j + zk and G(x, Y, z) = xi – yj+ zk. = -yi. Find V.(F × G) where F(x, y, z) = xi – -

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