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.

 

[blue_raver22]

Hello,

It's great to hear that you are learning about bases! To answer your question, there are actually different types of bases that exist in mathematics. The most common types are the null space basis and the range basis, which you mentioned. The null space basis refers to a set of vectors that span the null space of a matrix A, meaning that when these vectors are multiplied by A, the result will be the zero vector. The range basis, on the other hand, refers to a set of vectors that span the range of A, meaning that these vectors can be used to create any vector in the range of A through linear combinations.

Now, to answer your question about finding a basis for A, this typically refers to finding a basis for the column space of A. The column space of a matrix A is the span of its column vectors, and it is often denoted as Col(A). So, a basis for A would be a set of vectors that can be used to create any vector in Col(A) through linear combinations. This basis may or may not be the same as the null space basis or the range basis, as it depends on the specific properties of the matrix A.

I hope this helps clarify the different types of bases and their meanings. Keep up the great work in your studies!