Question

How do you wrap your head around the poorly written chapter 9.3 in 'Contemporary Linear Algebra'?

Answer 1

SOLUTION:

Here are some definitions are collected from different sources for chapter 9.3 in 'Contemporary Linear Algebra i.e. 'General Linear Transformations; Isomorphism'

1. Linear transformation :

Consider two linear spaces V and W. A function T from V to W is called a linear transformation

if: T(f + g) = T(f) + T(g) and T(kf) = kT(f) for all elements f and g of V and for all scalar k.

2. Rank, Nullity :

If the image of T is finite-dimensional, then dim(imT) is called the rank of T, and if the kernel of T is finite-dimensional, then dim(kerT) is the nullity of T.

If V is finite-dimensional, then the rank-nullity theorem holds

dim(V) = rank(T)+nullity(T) = dim(imT)+dim(kerT)

3. Isomorphisms and isomorphic spaces:

An invertible linear transformation is called an isomorphism. We say the linear space V and W are isomorphic if there is an isomorphism from V to W.

Properties of isomorphisms :

1. If T is an isomorphism, then so is T −1

2. A linear transformation T from V to W is an isomorphism if (and only if) ker(T) = {0}, im(T) = W

3. Consider an isomorphism T from V to W.If f1, f2, ...fn is a basis of V, then T(f1), T(f2), ...T(fn) is a basis of W.

4. If V and W are isomorphic and dim(V)=n, then dim(W)=n