Let V be a vector space over a field F, and let 3 be an ordered basis for V. Definethe transformation T: VF by T(v) [v]s. Show that T is linear.=

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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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Consider the vector space F of all functions from R to R with pointwise addition and scalar multiplication. (a) Show that the three vectors fi = e“, f2 = xe", f3 = x²e" are linearly independent in F. (b) Consider the subspace H = span{f1, f2, f3}. Let T : H → H be the linear transformation defined by df T(f)= + f. dx Find the matrix of T' with respect to the basis B = {fi, f2, f3} of H (i.e., find 8[T]B). (c) Is there a basis C of H in which c |T|c is diagonal? Explain why or why not.
Let V be a vector space, v, u € V, and let T₁: V → V and T2 VV be linear transformations such that T₁(v) = 6v 2u, T₁(u) = −2v + 4u, T₂ (v) = -2v 2u, T₂(u) = -7v+2u. Find the images of vand u under the composite of T₁ and T₂. (T₂T₁)(v) = (T₂T₁)(u) =
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Let V be a vector space, v, u € V, and let T₁ : V → V and T₂ : V → V be linear transformations such that T₁(u) = −5v + 3u, T₁ (v) = 5v +2u, T₂ (v) = −3v - 2u, T₂(u) = 5v – 2u. Find the images of v and u under the composite of T₁ and T₂. (T₂T₁)(v) = = (T₂T₁)(u) =
Let V be a vector space, v, u € V, and let T₁: V → V and T₂: V→ V be linear transformations such that Find the images of vand under the composite of T₁ and T₂. (T₂T₁)(v) = (T₂T₁)(u) = T₁(v) = 2v+3u, T₁(u) = -4v+4u, T₂(v) = -2v4u, T₂(u) = −7v+7u.
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Let V be a vector space over a field F, and let 3 be an ordered basis for V. Define the transformation T: V→ F" by T(v) = [v]s. Show that T is linear.