Understanding Matrix Ranks and Null Space: A Comprehensive Guide

In summary, the conversation revolved around the topic of matrices and the need for help with specific topics such as matrix rank and null space. The suggestion was made to search for resources on linear transformations and vector spaces, and links were provided for further reading.
  • #1
rdajunior95
25
0
Hi guys,

I basically need help with matrices, I know all the basics about them like inverse, determinants, eigenvalues and eigenvectors and all but I need help in some topics like matrix rank, null space and all.

I haven't read about them in any book so if you guys can post me links of some websites which explain these topics in detail please help me! :)
 
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  • #2
Then you are not really asking about "matrices", you are asking about "linear transformations" on vector spaces.

Google on "linear transformations", "vector spaces" or, more specifically, "null space", or, more generally, "linear algebra".
 
  • #3
Can someone please provide me with a link cause I searched before on google and could not find anything good!

So please help me :)
 
  • #4
Well, I googled on "vector space" "null space" and immediately got
http://online.redwoods.cc.ca.us/instruct/darnold/linalg/dim/dim.pdf
and
http://en.wikibooks.org/wiki/Linear_Algebra/Null_Spaces

They look good to me but I have no idea whether you would consider them "anything good".
 
Last edited by a moderator:
  • #5
thanks for the links!
 

Related to Understanding Matrix Ranks and Null Space: A Comprehensive Guide

1. What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It is also equal to the number of pivots in the row echelon form of the matrix.

2. How is the rank of a matrix related to its null space?

The rank-nullity theorem states that the sum of the rank and nullity of a matrix is equal to the number of columns in the matrix. This means that the rank and null space of a matrix are complementary to each other.

3. What is the difference between a full-rank and a rank-deficient matrix?

A full-rank matrix has a rank equal to the number of columns in the matrix, meaning that all of its columns are linearly independent. On the other hand, a rank-deficient matrix has a rank less than the number of columns, indicating that some of its columns are linearly dependent.

4. Can a matrix have a rank of 0?

Yes, a matrix can have a rank of 0 if all of its entries are 0. This means that all of the rows and columns of the matrix are linearly dependent, and the matrix only contains the zero vector in its null space.

5. How is the rank of a matrix useful in solving systems of linear equations?

The rank of a matrix can be used to determine if a system of linear equations has a unique solution, infinitely many solutions, or no solution. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has a unique solution. If the rank of the coefficient matrix is less than the rank of the augmented matrix, the system has infinitely many solutions. And if the rank of the coefficient matrix is less than the number of variables in the system, the system has no solution.

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