Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

410 VECTORS AND MATRICES [APP. A

Thesumofuandv, writtenu+v, is the vector obtained by adding corresponding components fromuandv;
that is,
u+v=(a 1 +b 1 ,a 2 +b 2 ,...,an+bn)

Thescalar productor, simply,product, of a scalarkand the vectoru, writtenku, is the vector obtained by
multiplying each component ofubyk; that is,

ku=(ka 1 ,ka 2 ,...,kan)

We also define
−u=− 1 (u) and u−v=u+(−v)

and we let 0 denote the zero vector. The vector−uis called thenegativeof the vectoru.

Thedot productorinner productof the above vectorsuandvis denoted and defined by

u·v=a 1 b 1 +a 2 b 2 +···+anbn

Thenormorlengthof the vectoruis denoted and defined by

‖u‖=

√

u·u=

√

a^21 +a^22 +···+an^2

We note that‖u‖=0 if and only ifu=0; otherwise‖u‖> 0.

EXAMPLE A.2 Letu=(2, 3,−4) andv=(1,−5, 8). Then

u+v=( 2 + 1 , 3 − 5 ,− 4 + 8 )=( 3 ,− 2 , 4 )

5 u=( 5 · 2 , 5 · 3 , 5 ·(− 4 ))=( 10 , 15 ,− 20 )

−v=− 1 ·( 1 ,− 5 , 8 )=(− 1 , 5 ,− 8 )

2 u− 3 v=( 4 , 6 ,− 8 )+(− 3 , 15 ,− 24 )=( 1 , 21 ,− 32 )

u·v= 2 · 1 + 3 ·(− 5 )+(− 4 )· 8 = 2 − 15 − 32 =− 45

‖u‖=

√

22 + 32 +(− 4 )^2 =

√

4 + 9 + 16 =

√

29

Vectors under the operations of vector addition and scalar multiplication have various properties, e.g.,

k(u+v)=ku+kv

wherekis a scalar anduandvare vectors. Many such properties appear in Theorem A.1, which also holds for
vectors since vectors may be viewed as a special case of matrices.

Column Vectors

Sometimes a list of numbers is written vertically rather than horizontally, and the list is called acolumn
vector. In this context, the above horizontally written vectors are calledrow vectors. The above operations for
row vectors are defined analogously for column vectors.

A.3Matrices

Amatrix Ais a rectangular array of numbers usually presented in the form

A=

⎡

⎢

⎢

⎣

a 11 a 12 ··· a 1 n

a 21 a 22 ··· a 2 n

...............................

am 1 am 2 ··· amn

⎤

⎥

⎥

⎦