In this exercise, we will outline a proof of the rank-nullity theorem: If T is a linear

Chapter 4, Problem 81

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In this exercise, we will outline a proof of the rank-nullity theorem: If T is a linear transformation from V to W, where V is finite dimensional, thendim( V) = dim(im T) + dim(ker T) = rank(7) + nullity(T).a. Explain why ker(7) and image (7) are finite dimensional. Hint: Use Exercises 4.1.54 and 4.1.57. Now consider a basis v\,..., vn of ker(7), where n = nullity(7), and a basis w\,...,wr of im(7), where r = rank(7). Consider elements u\.......ur in V such that 7(m;) = u)j for i = 1,..., r. Our goal is to show that the r + n elements u i,..., ur, i>i,..., vn form a basis of V; this will prove our claim. b. Show that the elements u i,..., ur, i>i,..., vn are linearly independent. Hint: Consider a relation ci u i + + crur + d\ v\ + + dn vn = 0, apply transformation T to both sides, and take it from there. c. Show that the elements u\,..., wr, U|____ vn span V. [Hint: Consider an arbitrary element v in V, and write T(v) = d\ w\ + + drwr . Now show that the element v d \u \-------- drur is in the kernel of 7, so that v d\u\ ... drur can be written as a linear combination of uj.......