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**[SOLVED] Linear Algebra - Direct Sums**

**1. Homework Statement**

Let W1, W2, K1, K2,..., Kp, M1, M2,..., Mq be subspaces of a vector space V such that

W1 = K1 [tex]\oplus[/tex]K2[tex]\oplus[/tex] ... [tex]\oplus[/tex]Kp

and

W2 = M1 [tex]\oplus[/tex]M2 [tex]\oplus[/tex]...[tex]\oplus[/tex]Mq

Prove that if W1 [tex]\cap[/tex]W2 = {0}, then W1 + W2 = W1 [tex]\oplus[/tex]W2 = K1 [tex]\oplus[/tex]K2[tex]\oplus[/tex]...[tex]\oplus[/tex] Kp [tex]\oplus[/tex] M1 [tex]\oplus[/tex]M2 [tex]\oplus[/tex]...[tex]\oplus[/tex]Mq

**3. The Attempt at a Solution**

Can we not just say W1 + W2 = W1 [tex]\oplus[/tex]W2 since their intersection is empty?

Then, by the definition of direct sum, the subspaces inside W1 and W2 cannot intersect each other.

Then can we say

W1 [tex]\oplus[/tex]W2 = K1 [tex]\oplus[/tex]K2[tex]\oplus[/tex]...[tex]\oplus[/tex] Kp [tex]\oplus[/tex] M1 [tex]\oplus[/tex]M2 [tex]\oplus[/tex]...[tex]\oplus[/tex]Mq ?