Proving that the Sum of Two Subspaces is a Subspace

Let U and W be subspaces of a finite-dimensional vector space V such that their intersection U ∩ W contains only the zero vector. We aim to prove that the direct sum U ⊕ W is a subspace of V. We will verify that U ⊕ W satisfies the three properties of a subspace: it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication. Let’s proceed step-by-step.

Step 1: Contains the Zero Vector

To show that the zero vector 0 is in U ⊕ W, we use the fact that both U and W contain the zero vector.

U contains the zero vector: 0 ∈ U

W contains the zero vector: 0 ∈ W

Therefore, we can write:

0 0 0 ∈ U ⊕ W

This confirms that U ⊕ W contains the zero vector.

Step 2: Closed Under Addition

To prove that U ⊕ W is closed under addition, we need to show that if x, y ∈ U ⊕ W, then x y ∈ U ⊕ W.

Let x, y ∈ U ⊕ W. By definition of the direct sum, we can express x and y as follows:

x u_1 w_1 for some u_1 ∈ U and w_1 ∈ W

y u_2 w_2 for some u_2 ∈ U and w_2 ∈ W

Now consider the sum x y:

x y (u_1 w_1) (u_2 w_2) (u_1 u_2) (w_1 w_2)

Since U and W are subspaces, they are closed under addition. Hence:

u_1 u_2 ∈ U

w_1 w_2 ∈ W

Therefore:

x y (u_1 u_2) (w_1 w_2) ∈ U ⊕ W

This shows that U ⊕ W is closed under addition.

Step 3: Closed Under Scalar Multiplication

To demonstrate that U ⊕ W is closed under scalar multiplication, we must show that if x ∈ U ⊕ W and c is a scalar, then c x ∈ U ⊕ W.

Let x ∈ U ⊕ W. Then we can write:

x u w for some u ∈ U and w ∈ W

Now consider the scalar multiplication:

c x c (u w) c u c w

Since U and W are subspaces, they are closed under scalar multiplication. Hence:

c u ∈ U

c w ∈ W

Thus:

c x c u c w ∈ U ⊕ W

This indicates that U ⊕ W is closed under scalar multiplication.

Conclusion

Since U ⊕ W contains the zero vector, is closed under addition, and is closed under scalar multiplication, we conclude that U ⊕ W is a subspace of V.