Question

linear sum w1+ w2 of two subspaces w1 and w2 of a vector space v (f) is subspace of v(f)​

Answer 1

Proof:

Let W1 and W2 are subspaces of vector space V(F) where F is any field and

Since 0 ∈ W1 and 0 ∈ W2 so 0 ∈ W1 + W2 and W1 + W2 is not empty

Suppose

x and y ∈ W1 + W2

then

x = u1 + v1        for some u1 ∈ W1 and v1 ∈ W2

Similarly

y = u2 + v2        for some u2 ∈ W1 and v2 ∈ W2

Now

x + y = u1 + v1 + u2 + v2

        = u1 + u2 + v1 + v2

        = u3 + v3          ( ∵ W1 and W2 are subspaces  So u1 + u2 = u3 for u3 ∈ W1 and similarly v1 + v2 = v3 for some v3 ∈ W2 )

Thus

x + y = u3 + v3 ∈ W1 + W2

Hence closure law of addition holds in W1 + W2

Now

Let

α ∈ F then every u + v ∈ W1 + W2

α(u + v)  = αu + αv

             = u° + v° ∈ W1 + W2        ( ∵ W1 and W2 are subspaces so  αu = u° and αv = v° )

It shows that scalar multiplication holds in W1 + W2 so

From first and second property

W1 + W2 is the subspace of  V(F) where F is the field

Answer 2

Answer:

If W1, W2 are disjoint subspaces of vector space then