subspace of a Linear Transformation?

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December 15th 2012, 06:14 AM
alphaknight61

subspace 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.
December 15th 2012, 07:21 AM
jakncoke

Re: subspace of a Linear Transformation?

3 Conditions must hold

1) $0_{w} \text{ must be in } L(V_1)$
Well since its a linear transformation $L(x - x) = L(x) - L(x) = 0_{w}$ , so $L(0_{v}) = 0_{w}$

2) closed under addition. If $x_1, x_2 \in L(V_1)$ then $x_1 + x_2 \text{ must also be in } L(V_1)$
Pf: Take $x_1, x_2 \in L(V_1)$ , so $\exists v_1, v_2 \in V \text{ such that } L(v_1) = x_1 \text{ and } L(v_2) = x_2$ . So again, linear transformation, $L(v_1) + L(v_2) = L(v_1 + v_2) = x_1 + x_2 \text{ since } v_1+ v_2 \in V$ , and so $x_1 + x_2 \in L(V_1)$

3)closed under taking scalar product.
Pf: Take $c \in \mathbb{F}$ , then for any $x \in L(V_1) \exists v \in V \text{ such that } L(v) = x \text{ and so } c * x = c * L(v) = L(cv)$ , since $cv \in V$ , $cx \in L(V_1)$
December 15th 2012, 10:45 AM
HallsofIvy

Re: 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.