Let L: V--> W be a Linear Transformation from a vector space V into W. The image of the subspace V1 of V is: L(V1) ={β ϵ W | Ǝ α ϵ V where β = L(α)}. Show that L (V1) is a subspace of W.
3 Conditions must hold
1)
Well since its a linear transformation , so
2) closed under addition. If then
Pf: Take , so . So again, linear transformation, , and so
3)closed under taking scalar product.
Pf: Take , then for any , since ,
Note: the first of jakncoke's requirement is, in some books, change to "the set is non-empty. Obviously if it contains the 0 vector, as jakncoke requires, it is non-empty. Conversely, if a nonempty set satisfies the other two requirements, since it is non-empty, there exist some vector, v. Because it satisifies (3), it also contains (-1)v= -v. And because it satisfies (2), v+ (-v)= 0 is in the set.