Let V, W be finite dimensional vector spaces over F and let T : V → W be alinear map. Recall the isomorphismÞy : V → V** by sending v toevv. Prove that the following diagram commutesVie, that w o T = T** o OyÞvWΦνΦw→ V**T**→ W**(Hint: Recall that T** : V** → W** sends a linear functional: V* → F to the linear functional poT*: W* → F. That is T** (y) = yoT* € W**. You willthen evaluate what this is on a linear functional y € W* )

First we list all the maps we have with definitions. Here we have given that T:V→W we have map…

Let V, W be finite dimensional vector spaces over F and let T: VW be a linear map. Recall the isomorphism V** by sending v to Φv:V → ev. Prove that the following diagram commutes V T W Φρ Φw → V** → W** O ie, that w o T = T** ○ Þy (Hint: Recall that T** : V** → W** sends a linear functional 4: V* → F to the linear functional yoT* : W* → F. That is T** (y) = y0T* € W**. You will then evaluate what this is on a linear functional y € W* )

Let V, W be finite dimensional vector spaces over F and let T: VW be a linear map. Recall the isomorphism V by sending v to Φv:V → ev. Prove that the following diagram commutes V T W Φρ Φw → V → W O ie, that w o T = T ○ Þy (Hint: Recall that T : V → W** sends a linear functional 4: V* → F to the linear functional yoT* : W* → F. That is T** (y) = y0T* € W**. You will then evaluate what this is on a linear functional y € W* )

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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