Originally Posted by

**adambuck** Your basis would be what you have written: [1,6;0,5] and [0,0;1,0].

Every element of the set U = {[a,6a;b,5a] : a,b are real} can always be written as a linear combination of the two elements. Since it can be written uniquely in this way, it is a basis. Does that make sense? If W = {[a+b, b; 2a+b, 3a] : a,b are real}, a basis would be {[1,0; 2, 3], [1,1; 1,0]}. The comment you listed above is saying that {[1,0; 0,0], [0,1; 0,0], [0,0; 1,0], [0,0; 0,1]} is a basis for the space of all 2x2 matrices, but there are many more bases for this set.