Vector space/Example II

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 C^mxn.

So Example I would be denoted R^1xn, 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.