Difference among types of bases.

[smithnya]

Hello everyone,

I am currently learning about bases, and I am confused about different types of bases. I understand how to obtain a basis for the null space of A or a basis for the range of A. I recently ran into a problem that simply asked me to find a basis for A. What does this mean? Does it mean any basis (for N(A) or R(A))? OR is there such a thing as a separate basis for A not having to do with R(A) or N(A)?

[Sina]

Maybe it wants you to visualize the set of matrices as a vector space?

Consider for example the set of matrices of the form

|a 0|
|0 b|

They are spanned by the two independent elements

|1 0|
|0 0|

|0 0|
|0 1|

so this is a basis for the subspace of matrices given above. What is the exact form of A and of the question?

[homeomorphic]

If A is a matrix, the question is nonsense.

If A is a vector space, then it makes sense.

[Deveno]

my guess is what is meant is a basis for the column-space or row-space of A.