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.

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- Dec 15th 2012, 07:14 AMalphaknight61subspace of a Linear Transformation?
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.

- Dec 15th 2012, 08:21 AMjakncokeRe: subspace of a Linear Transformation?
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 , - Dec 15th 2012, 11:45 AMHallsofIvyRe: subspace of a Linear Transformation?
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.