crypt50 said:Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help would be greatly appreciated.
A quotient space is a mathematical concept that arises in the field of linear algebra. It is the set of all possible cosets of a given vector space modulo a specific subspace. In simpler terms, it is the set of all possible ways to divide a vector space into smaller subspaces.
In this statement, U refers to a specific subspace of the vector space R^n. This subspace is used to create the quotient space R^n/U.
In this context, isomorphism refers to a bijective linear transformation between two vector spaces. This means that there is a one-to-one correspondence between the elements of the two vector spaces, and the linear structure and operations are preserved.
When two spaces are isomorphic, it means that they are essentially the same, in the sense that they have the same structure and properties. In the context of this statement, it means that the quotient space R^n/U has the same structure and properties as the subspace W.
The isomorphism between R^n/U and W is proven by showing that there exists a bijective linear transformation between the two spaces. This can be done by constructing a specific linear transformation that maps the elements of R^n/U to the elements of W, and then showing that this transformation is bijective.