Chapter 1

Review of Some Linear AlgebraThis material can be referred to as needed. It is here in order to make the book self con-tained.

1.1 The Matrix of a Linear MapRecall the definition of a linear map. First of all, these need to be defined on a linear spaceand have values in a linear space.

Definition 1.1.1 Let T : V →W be a function. Here V and W are linear spaces.Then T ∈L (V,W ) and is a linear map means that for α,β scalars and v1,v2 vectors,

T (αv1 +βv2) = αT v1 +βT v2

Also recall from linear algebra that if you have T ∈L (Fn,Fm) it can always be un-derstood in terms of a matrix. That is, there exists an m× n matrix A such that for allx ∈ Fn,

Ax= Tx

Recall that, from the way we multiply matrices,

A =(

Te1 · · · Ten)

That is, the ith column is just Tei.

1.2 Block Multiplication of MatricesConsider the following problem(

A BC D

)(E FG H

).

You know how to do this. You get(AE +BG AF +BHCE +DG CF +DH

).

Now what if instead of numbers, the entries, A,B,C,D,E,F,G are matrices of a size suchthat the multiplications and additions needed in the above formula all make sense. Wouldthe formula be true in this case?

Suppose A is a matrix of the form

A =

 A11 · · · A1m...

. . ....

Ar1 · · · Arm

 (1.1)

where Ai j is a si× p j matrix where si is constant for j = 1, · · · ,m for each i = 1, · · · ,r. Sucha matrix is called a block matrix, also a partitioned matrix. How do you get the block

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