Matrix rank math help

  • Thread starter physicsss
  • Start date
  • #1
physicsss
319
0
Suppose that a matrix A is formed by taking n vectors from R^m as its columns.

a) if these vectors are linearly independent, what is the rank of A and what is the relationship between m and n?

is the rank the same as the dimension of the column space, or n, and m less than or equal to n?

b) if these vectors span R^m instead, what is the rank of A and what is the relationship between m and n?

is the rank m and m=n?

c) if these vectors form a basis for R^m, what is the relationship between m and n then?

is m=n?
 

Answers and Replies

  • #2
lurflurf
Homework Helper
2,450
148
physicsss said:
Suppose that a matrix A is formed by taking n vectors from R^m as its columns.

a) if these vectors are linearly independent, what is the rank of A and what is the relationship between m and n?

is the rank the same as the dimension of the column space, or n, and m less than or equal to n?

b) if these vectors span R^m instead, what is the rank of A and what is the relationship between m and n?

is the rank m and m=n?

c) if these vectors form a basis for R^m, what is the relationship between m and n then?

is m=n?
A:R^n->R^m
a)
The first part is easy to see rank(A)=n=dim(span(colums))=coulumspace
as you said but n<=m as it must be. If m<n R^m has a linearly independent basis with more than m vectors, but this cannot be as we know the standard basis has m vectors and for finite diminsional vector spaces two linearly independent spaning sets have the same finite number of elements.
b)
That is right
c)
this is also right.
 

Suggested for: Matrix rank math help

Replies
4
Views
4K
  • Last Post
Replies
17
Views
6K
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
10
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
18
Views
2K
  • Last Post
Replies
1
Views
814
Top