Understanding Scalar Transformations in Matrix Dimensions

  • Thread starter the_kid
  • Start date
In summary, the conversation discusses the relationship between the dimensions of a matrix with column vectors that may or may not be a scalar transformation of each other. The question of what can be said about the dimensions is raised, but there is uncertainty surrounding the definition of a "scalar transformation." It is suggested that if it means a scalar multiple, then no conclusions can be drawn, but if it means a linear combination with n-1 independent columns, then the column rank is n-1. It is also noted that the row rank must be n-1 as well, meaning there must be at least n-1 rows, so m must be greater than or equal to n-1. The lack of clarity in the question is identified as a potential
  • #1
the_kid
116
0
Just a quick question I've been giving some thought to today. Suppose A is an mxn matrix with column vectors such that only one of them is a scalar transformation of another. What can we say about the relationship of m and n?
 
Physics news on Phys.org
  • #2
No thoughts?
 
  • #3
what is a "scalar transformation"? if it means scalar multiple, then i don't see right off how to say anything. if it means linear combination, in the sense that there are n-1 independent columns, then it say the column rank is n-1. since then the row rank must also be n-1, we know there are at least n-1 rows, so m ≥ n-1.

your problem in receiving no answers is your unclear statement of the question.
 

1. What is the nullity-rank question?

The nullity-rank question is a concept in linear algebra that asks about the relationship between the nullity (dimension of the null space) and the rank (dimension of the column space) of a matrix. It is often used to understand the linear independence of a set of vectors or the solutions to a system of linear equations.

2. How is the nullity-rank question related to matrices?

The nullity-rank question is directly related to matrices because it is typically asked in the context of a matrix representing a linear transformation. By considering the nullity and rank of the matrix, we can determine important properties of the transformation, such as whether it is invertible or if it has a unique solution.

3. Can the nullity-rank question be applied to non-square matrices?

Yes, the nullity-rank question can be applied to non-square matrices. However, in this case, the nullity and rank are not necessarily equal and the question becomes more complex. For non-square matrices, the nullity is defined as the dimension of the null space of the matrix, while the rank is defined as the maximum number of linearly independent columns (or rows) of the matrix.

4. How can the nullity-rank question be used in practical applications?

The nullity-rank question has many practical applications in fields such as engineering, physics, and computer science. For example, it can be used to determine the linear independence of a set of vectors in a vector space, to analyze the stability of a control system, or to compress data in image and signal processing.

5. Is there a formula for solving the nullity-rank question?

Yes, there is a formula for solving the nullity-rank question. For a square matrix A with dimensions m x m, the nullity and rank are related by the formula: nullity(A) + rank(A) = m. This means that if we know the value of one of these quantities, we can easily calculate the other. However, for non-square matrices, the formula is more complex and depends on the specific properties of the matrix.

Similar threads

  • Linear and Abstract Algebra
Replies
4
Views
766
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
993
Replies
12
Views
3K
Replies
27
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
919
  • Linear and Abstract Algebra
Replies
12
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
487
  • Linear and Abstract Algebra
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
951
Back
Top