Matrix rank math help

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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?
 

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  • #2
lurflurf
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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.
 

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