2. Let V and W be vector spaces over a field F and let L: V → W be a linear transformation. Assume that dim(V) = n for some n e N. (a) Define ker(L), the kernel of L and im(L), the image of L. (b) Prove that ker(L) is a subspace of V. (c) If the rank of L is r, what is the dimension of the kernel of L? (d) Give an example of V, W and L where the rank r of L satisfies r = 0. (You can choose a value for n.)

2. Let V and W be vector spaces over a field F and let L: V → W be a linear transformation. Assume that dim(V) = n for some n e N. (a) Define ker(L), the kernel of L and im(L), the image of L. (b) Prove that ker(L) is a subspace of V. (c) If the rank of L is r, what is the dimension of the kernel of L? (d) Give an example of V, W and L where the rank r of L satisfies r = 0. (You can choose a value for n.)

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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