Let V be a vector space, and suppose that U and W are both subspaces of...

Let V be a vector space, and suppose that U and W are both subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a subspace of V.

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Expert Solution

Defination : Let V be a vector space , a subset A V is said to be a subspace of V if the following two conditions are hold

(i) x € A and y € A then x +y € A

(ii) x € A and c be a constant then cx € A.

Now given U and W both are subspace of V .

(i) Let x € U W .

Then x € U and x€ W . Now since both U and W are subspace of V so x + y belongs to both U and V .

This implies x + y € U W . So x € U W and y € U W implies x + y belongs to U W .

So condition (i) satisfies .

(ii) Suppose x € U W and c be a scalar .

Since x € U W so x € U and x € W .

Now as both U and W are subspace of V then ,

cx € U and cx € Y => cx € U W .

So x € U W and c be a scalar implies cx € U W .

So condition (ii) satisfies .

Hence U W is a subspace of V .

Colby Messinger answered 1 year ago

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