I have a question about how to find a basis for a subspace in (Linear Algebra) ,like how to find a basis for the kernel?And how to find a basis for the range?

Question

Or finding a basis for any subspace , like .... can I find a basis for any subspace or does it have some kinda conditions ? It's really confusing me ...

Answer 1

There are four fundamental spaces to a matrix all stemming from the equations Ax=0

You will have

Column space, which is also the range or image.

Row space

Left null space, which comes from A^tx=0

Right null space, which is also the kernel and is the vectors from the vector solution to Ax=0 for nontrivial infinite solutions. (trivial means x=0 in linear algebra).

The process is quite simple really, all you need to do is rref( [ax|0] ) and then identify pivots and extract vectors. If you would like to see how it is done, feel free to contact me. I offer are free 1-hour trial/get to know one another session for students that are interested regular tutoring.

JJtheTutor