show that V=M where m is m*n matrix of real entries is a vector space

Question

show that V=M_m×n is a m×n matrix of real entries is  a vector space

Answer 1

To be a vector space, we need to check 3 things:

 

1.  0 exists in V

2.  If A, B exist in V, A+B exists in V

3.  If A exists in V, cA exists in V

 

1.  Since 0 is a real number, clearly a matrix 0 consisting of all 0's exists in V

 

2.  Let A, B exist in V.  Then A and B are (m x n) matrices with real entries.  Therefore A + B will also be (m x n) and will also have real entries.  This implies that A + B exists in V

 

3.  Let c be a real number and let A exist in V.  Then A is (m x n) with real entries.  Therefore cA will also be (m x n) and will also have real entries.  This implies that cA exists in V

 

Therefore V is a vector space.

 

 

J.T.