Principles of Mathematics in Operations Research

2.2 Linear transformations, matrices and change of basis 17

Remark 2.1.16 Since Span(X) C V, in order to show that it covers V, we

only need to prove that Vv € V, v € Span(X).

Example 2.1.17 In Kn, E = {ej}"=1 is a basis since E is linearly indepen-

dent andVa = (ai,a 2 ,.. -,an)T € Kn, a = a^ei -\ 1- ane„ € Span(E).

X — {xi}™=1 is also a basis for Rn since Va = (ai,a2,... ,an)T € Rn,

a = aixi + (a 2 - "l)^ H 1- K - an-i)xn £ Span(X).

Proposition 2.1.18 Suppose X = {a?i}7=i *s a ^0Sl'^5 /or ^ ower ^- ^en»

aj Vw £ l^ can be expressed as v = E?=i aixi where cti 's are unique.

b) Any linearly independent set with exactly n elements forms a basis.

c) All bases for V contain n vectors, where n is the dimension ofV.

Remark 2.1.19 Any vector space V of dimension n and an n-dimensional

field F™ have an isomorphism.

Proof. Suppose X = {xi}"=1 is a basis for V over F. Then,

a) Suppose v has two different representations: v = Y17=iaix' = Y^i=i&iXi-

Then, 6 — v — v = Ei=i(ai ~ Pi)xi =^ °-% — ft, Vz — 1,2,..., n. Contra-

diction, since X is independent.

b) Let Y = {j/i}7=i be linearly independent. Then, yi = Yl^ixi (40> where at

least one S{ ^ 0. Without loss of generality, we may assume that Si ^ 0.

Consider Xi = {yi,x?,... ,xn}. Xi is linearly independent since 6 =

fttfi+E?=2#** = /MEW**+Er= 2 ft^ = ft^^i+£r= 2 (ft^ +

fr)xi =*• ft<5i = 0; PiSi + ft = 0, Vi = 2,..., n =*• )8i = 0 (<Ji # 0); and

ft = 0, Vi = 2,..., n. Any oeK can be expressed as v = E?=i 7*:c* =

7iai + E_r= 2 7iffi

u = 7i(<^r

1

2/i - Er=2

<J

r

1

^

a;

i)

(

*

)

= (7i^r

1

)yi + E"= 2 (7i - ns^s^.

Thus, Span(Xi) = V.

Similarly,

X2 = {yi,y2,x 3 ,...,xn} is a basis.

Xn = {2/1,2/2, • • • ,2/n} = Y is a basis.

c) Obvious from part b). •

Remark 2.1.20 Since bases for V are not unique, the same vector may have
different representations with respect to different bases. The aim here is to
find the best (simplest) representation.

2.2 Linear transformations, matrices and change of basis

2.2.1 Matrix multiplication

Let us examine another operation on matrices, matrix multiplication, with
the help of a small example. Let A e K3x4, B G R4x2, C € R3x2