Pseudoinverse - change of basis?

[kviksand81]

Hello,

I was wondering if the pseudoinverse can be considered a change of basis?

If an m x n matrix with m < n and rank m and you wish to solve the system Ax = b, the solution would hold an infinite number of solutions; hence you form the pseudoinverse by A^T(A*A^T)^-1 and solve for x to get the minimum norm solution. And since A was the original basis the pseudoinverse must be a new basis...? Or am I getting it all wrong?

 

[HallsofIvy]

I don't understand what you mean by "since A was the original basis the pseudoinverse must be a new basis". A matrix is NOT a "basis". A basis is a collection of vectors, not a matrix or linear transformation. A linear transformation is represented by a matrix using a specific basis but finding the generalized inverse does not change the basis.

 

[kviksand81]

Sorry. From the original problem where m < n, there is not a set of linearly independent vectors to yield a exact solution to the system Ax = b. Then a common technique to solve for a vector that comes as close as possible, would be to form the pseudoinverse, right? This pseudoinverse, is that a new basis or what is it?

Hope that it is more clear what I mean now! :-)

 

[HallsofIvy]

No, a basis, for a given vector space, is a set of vectors that span the vector space and are independent so what you are describing is not a basis.

 

[kviksand81]

Ok.

If I then solve the system Ax =b where the matrix A (m x n) with rank m by means of the right-hand pseudo inverse A^T(A*A^T)^-1, what I get is the minimum norm solution to the m independent columns of A? Or...?

What happens to the remaining free variables? Are they in the null-space?