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Direct Sum
This 6-minute video by Lorenzo Sadun explained direct sums to me, after I'd seen the word used many times and wasn't really getting it. The idea is that if you wanted to combine two incompatible things, you could pad each of them with zeroes.
Example:
(a₁, a₂, a₃) + (b₁, b₂, b₃)
(a₁, a₂, a₃, 0, 0, 0) + (0, 0, 0, b₁, b₂, b₃)
(a₁, a₂, a₃, b₁, b₂, b₃)
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