# Span, Basis, Dimension

jlucas134
I need some help to understand the concepts of span, basis, and dimension.

1-How do you determine if a set of vectors [in matrix form] span a region?
-Do you set the given matrix set to arbitary numbers and see if
there is a unique, infinite, or no solution?
ie, set a 4X4 matrix = to [a b c d] and determine the type of solution?

2. How do you find a basis for the kernel of the linear map L : R^4 goes to
R4 corresponding to multiplication of a given matrix?
-Do you do the same thing as above, but set the given matrix equal to
zero to find the kernal and after you find the kernal, do you find the
basis?

3. How do you find a basis for the range of this same map?
Do you find the range, then the basis?

I think if I understand these concepts, I can do the homework. Any assistance would be appreciated.

Homework Helper
I need some help to understand the concepts of span, basis, and dimension.

1-How do you determine if a set of vectors [in matrix form] span a region?
-Do you set the given matrix set to arbitary numbers and see if
there is a unique, infinite, or no solution?
ie, set a 4X4 matrix = to [a b c d] and determine the type of solution?

Yes, you do. Note that the solution (i.e. representation) doesn't have to be unique if you're talking about a spanning set. Existance is important.

2. How do you find a basis for the kernel of the linear map L : R^4 goes to
R4 corresponding to multiplication of a given matrix?
-Do you do the same thing as above, but set the given matrix equal to
zero to find the kernal and after you find the kernal, do you find the
basis?

Of course, you first have to find the kernel, i.e. you have to know what a set looks like in order to do anything with it.

Homework Helper
these questions are answered in every linear algebra text. indeed this is about all there is to basic concrete linear algebra. do you have a book? e.g. shifrin and adams?

Last edited: