Example II: Let M be the set of all (mxn) matrices, with complex numbers as entries. Let C be the field of complex numbers. Then if
P is in M, P= |p11 p12 p13...p1n|
|p21 p22 p23...p2n|
|p31 p32 p33...p3n|
|.......................|
|.......................|
|pm1 pm2 pm3...pmn|
- where pij is in C.
Define vector addition in M:
P+Q= |p11 p12 p13...p1n| |q11 q12 q13...q1n| |p21 p22 p23...p2n| |q21 q22 q23...q2n| |p31 p32 p33...p3n| |q31 q32 q33...q3n| = | . | + | . | | . | | . | |pm1 pm2 pm23 pmn| |qm1 qm2 qm3...qmn|
|p11+q11 p12+q12 p13+q13...p1n+q1n| |p21+q21 p22+q22 p23+q23...p2n+q2n| |p31+q31 p32+q32 p33+q33...p3n+q3n| |. | |. | |pm1+qm1 pm2+qm2 pm3+qm3...pmn+qmn|
Define scalar multiplication:
|p11 p12 p13...p1n| |c*p11 c*p12 c*p13...c*p1n|
|p21 p22 p23...p2n| |c*p21 c*p22 c*p23...c*p2n|
c* |p31 p32 p33...p3n| |c*p31 c*p32 c*p33...c*p3n|
| . | = | |
| . | | |
|pm1 pm2 pm3...pmn| |c*pm1 c*pm2 c*pm3...c*pmn|
Then M is a vector space over C and we denote this as Cmxn.
So Example I would be denoted R1xn, or more simply, Rn.
In analysis, many function sets have the structure of a vector space. In analysis, a vector space is called a linear space.
- See also : Vector space
