Question

Let U and W be subspaces of F^n and let T: F^n → F^n be a linear transformation. Prove/disprove:

(a) For all w in W, if v = T(w), then v is in U.

Answer 1

Without further details on the relationship between the linear transformation T and the subspace U, one cannot conclude that the image of any vector w in W under T will belong to U.

Step-by-step explanation:

The question is asking whether the image of a vector w in subspace W under a linear transformation T will always be in another subspace U. This statement can neither be proven nor disproven without additional information about the relationship between T and U. Specifically, if T is a linear transformation such that T(W) is contained in U, then indeed every v = T(w) for w in W will be in U. However, in the absence of such a condition, there's no guarantee that v will be in U just because it is the image of a vector in W under T.