Understanding Span, Basis, and Dimension: A Guide to Linear Algebra Concepts

[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.

Thanks in advance!

 

[radou]

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.

 

[mathwonk]

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?

 

[mathwonk]

use gaussian elimination.

given a set of vectors in R^n, put them in as either rows or columns in a matrix and row reduce. they did span iff the number of non zeroes rows remaining is n, iff the number of "pivot" columns is n.

to find a basis for the image of a matrix, reduce and take the pivot columns as your basis. use the usual solution procedure on the reduced matrix (back substitution) to write down a basis of the null space or kernel.

to see if a given vector lies in the span of the columns of a matrix, add it as another column, reduce and see if this new last column is a pivot, (if so it is not in the span).