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2. Let V and W be vector spaces and let TE L(V, W). (a) Suppose V is finite-dimesnional, m is a finite positive integer, and wi,..., Wm is a basis for range T. Show that there exist vectors v, e V, 1

2. Let V and W be vector spaces and let TE L(V, W). (a) Suppose V is finite-dimesnional, m is a finite positive integer, and wi,..., Wm is a basis for range T. Show that there exist vectors v, e V, 1

Advanced Engineering Mathematics
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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2. Let V and W be vector spaces and let T E L(V, W). (a) Suppose V is finite-dimesnional, m is a finite positive integer, and wi,..., Wm is a basis for range T. Show that there exist vectors v; e V, 1 <i< m, such that Tu; = wi, and show that these vectors v; are independent. Also show that span(v1,..., vm) O null T = V. (b) If V is infinite-dimensional (which means m may also be infinite), does the result of (a) still hold? Explain.
Transcribed Image Text:2. Let V and W be vector spaces and let T E L(V, W). (a) Suppose V is finite-dimesnional, m is a finite positive integer, and wi,..., Wm is a basis for range T. Show that there exist vectors v; e V, 1 <i< m, such that Tu; = wi, and show that these vectors v; are independent. Also show that span(v1,..., vm) O null T = V. (b) If V is infinite-dimensional (which means m may also be infinite), does the result of (a) still hold? Explain.
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