Linear Algegra: Nul A and Col A relationship if Nul A is not the zero space

In summary, if R is a 6X6 matrix and Nul R is not the zero space, then the dimension of the column space of R is equal to 6. This is known as the rank-nullity theorem. Additionally, if the nullity of R is greater than 0, then the rank of R will be less than 6.
  • #1
imagenesis
1
0
1. If R is a 6X6 matrix, and Nul R is not the zero space, what can you say about Col R?





3. Well we know that there is some vector that can be added to R to form 0... But how does that relate to Col R ? I mean I really don't know. I am guessing it has something to do with the m where mxn is the size of R.
 
Physics news on Phys.org
  • #2
Look at the rank-nullity theorem.
 
  • #3
If R is a 6 by 6 matrix- i.e. represents a linear transformation from 6 dimensional vector space to a 6 dimensional vector space, then the dim of the null space of R plus the dimension of the column space of R is equal to 6- that the rank-nullity theorem Vid mentioned. If the nullity of R is greater than 0, what can you way about the rank of R?
 

1. What is the relationship between the null space and column space if the null space is not the zero space?

The null space and column space are both fundamental concepts in linear algebra. The null space of a matrix A, denoted as Nul A, is the set of all vectors x that satisfy the equation Ax = 0. The column space of a matrix A, denoted as Col A, is the span of the columns of A. If the null space of A is not the zero space, then it means that there exists at least one non-zero vector that satisfies Ax = 0.

2. How are the dimensions of the null space and column space related if the null space is not the zero space?

If the null space of a matrix A is not the zero space, then the dimension of the null space, denoted as dim(Nul A), will be greater than 0. This means that there is at least one free variable in the reduced row echelon form of A. The rank-nullity theorem states that the sum of the dimensions of the null space and column space is equal to the number of columns in A. Therefore, if dim(Nul A) > 0, then the dimension of the column space, denoted as dim(Col A), will be less than the number of columns in A.

3. How does the existence of a non-zero null space affect the invertibility of a matrix?

If the null space of a matrix A is not the zero space, then it means that A is not a one-to-one (injective) mapping. This also means that A does not have a unique inverse, as there are multiple vectors that can map to the zero vector. Therefore, a matrix with a non-zero null space is not invertible.

4. Can the null space and column space of a matrix be equal if the null space is not the zero space?

No, the null space and column space of a matrix A cannot be equal if the null space is not the zero space. This is because the column space is always a subspace of the vector space in which the null space resides. If the null space is not the zero space, then it means that there is at least one non-zero vector that satisfies Ax = 0. This vector cannot be in the column space, as the column space only contains the linear combinations of the columns of A.

5. How can the existence of a non-zero null space impact the solutions of a system of equations represented by a matrix?

If the null space of a matrix A is not the zero space, then it means that there are infinitely many solutions to the system of equations Ax = 0. This is because the null space contains all the solutions to the homogeneous system of equations Ax = 0. In other words, the existence of a non-zero null space means that there are more solutions to the system of equations than just the trivial solution (x = 0).

Similar threads

Help with linear algebra: vectorspace and subspace
    • Calculus and Beyond Homework Help
Replies
15
Views
2K
Does the Null Space of a 2x3 Matrix Determine its Column Space?
    • Calculus and Beyond Homework Help
Replies
15
Views
2K
Describe all vectors orthogonal to col(A) with a twist
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Null space vs Col space dimension?
    • Calculus and Beyond Homework Help
Replies
2
Views
5K
Direct Proof that every zero of p(T) is an eigenvalue of T
    • Calculus and Beyond Homework Help
Replies
24
Views
795
Determining value of r that makes the matrix linearly dependent
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Linearly independent functions with identically zero Wronskian
    • Calculus and Beyond Homework Help
Replies
1
Views
278
Linear algebra rank and dimensions
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
Null space of dual map of T = annihilator of range T
    • Calculus and Beyond Homework Help
Replies
0
Views
449
I Vector visualization of multicollinearity
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
623

Hot Threads

Recent Insights

Back
Top