How Does Matrix Rank Relate to Vector Independence and Dimensionality?

[physicsss]

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?

 

[lurflurf]

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.

 

[thepatientmental]

a) If the n vectors are linearly independent, the rank of A would be n. In this case, the dimension of the column space would also be n. The relationship between m and n would be that m is equal to or greater than n, as the vectors are taken from R^m.

b) If the n vectors span R^m, the rank of A would be m. In this case, m would be equal to n, as the vectors are taken from R^m and are able to span the entire space. Therefore, the relationship between m and n would be that m is equal to n.

c) If the n vectors form a basis for R^m, then m would also be equal to n. This is because a basis consists of linearly independent vectors that span the entire space. Therefore, the relationship between m and n would be that m is equal to n.