Solution

This is a problem which looks horrible, but is actually easy! Call the first column and the second one . Then note that is the column of . Call this . Now, the columns are . So, clearly span the column space and are actually in the column space as well. So the basis is , if they are independent!

It is easy to see that they are independent! Why?

If they were dependent, then will be a multiple of , but all multiples of have equal entries and does not satisfy this property!

Warning: This meaning of linear dependence only works when dealing with two vectors! Don't use it for three or more vectors. Thus any three columns of our matrix are dependent, even though no one columns is a multiple of another:

Alternatively one can row reduce the matrix to get and subtract (n+1) times row 2 from row 1 to get . The first two columns of this matrix are a basis for its column space so the first two columns of the original matrix are a basis for it.