Linear Algebra - Basis

Question

Show that S = [1 2 0 0] , [1 0 2 0] , [1 0 0 2] , [2 1 0 0] is a basis for V = M2*2(R) , R=set of real numbers

Answer 1

1 2 0 0

1 0 2 0

1 0 0 2

2 1 0 0

-2 * row 1 + row4

---------------------

1 2 0 0

1 0 2 0

1 0 0 2

0 -3 0 0

row4/-3

--------------

1 2 0 0

1 0 2 0

1 0 0 2

0 1 0 0

-2*row4 + row1

---------------

1 0 0 0

1 0 2 0

1 0 0 2

0 1 0 0

-row1 + row3

--------------

1 0 0 0

1 0 2 0

0 0 0 2

0 1 0 0

row3/2

--------------

1 0 0 0

1 0 2 0

0 0 0 1

0 1 0 0

-row1 + row2

--------------

1 0 0 0

0 0 2 0

0 0 0 1

0 1 0 0

row2/2

-------

1 0 0 0

0 0 1 0

0 0 0 1

0 1 0 0

swapping rows gives

the identity matrix