What makes an equation linear?

[joshuacook]

from @giladgressel on slack

[joshuacook]

I think that there will be a lot more discussion about this. We will come to define a vector space as having the following two properties:

1. closed under vector addition (if u,v in V, then u+v in V)
2. closed under scalar multiplication (if a in R and u in V, then a*u in V)

If you look at wikipedia's page on linearity this is very similar to the definition of linearity:

In mathematics, a linear map or linear function f(x) is a function that satisfies the following two properties:
Additivity: f(x + y) = f(x) + f(y).
Homogeneity of degree 1: f(αx) = αf(x) for all α.

[mxbi]

closed under vector addition (if u,v in V, then u+v in V)

What exactly does the word 'closed' mean here?

[joshuacook]

The statement in the parentheses is identical. "Closed" means that the sum of the two vectors is in the same vector space as the original two vectors.

For multiplication it means that any scalar multiple of a vector is in the same vector space as the original vector.

[joshuacook]

So I guess to say simply, "closed" means "in the same vector space".