EMMETT09
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What does it means when a vector is closed under addition and scalar Multiplication?
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Smaug123
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(Original post by EMMETT09)
What does it means when a vector is closed under addition and scalar Multiplication?
A vector can't be closed under addition and scalar multiplication. A vector <space> can be (and, indeed, is). It means that adding two vectors in the space gives you another vector in the space, and multiplying a vector by a scalar gives you a vector in the space.
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reyjusuf
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if v1 and v2 is in the vector space, then v1+v2 has to be in that vector space and cv1 has to be in that vector space.

for example is the set { c(x,y); x = y ; c =) R } a vector space?

this vector space consists of scalar multiples of the vector (1,1).

is (1,1)+(1,1) in the vector space? yes, because (2,2) is in the vector space. => close under addition
is 5(1,1) in the vector space? yes, because (5,5) is in the vector space. => close under scalar multiplication
is (0,0) in the vector space? yes. because 0(1,1) = (0,0) is in the vector space

therefore the set { c(x,y); x = y ; c =) R } is a valid vector space

however, the set { c(x,y); x = y; x,y >= 0 ; c =) R } is not a vector space

because -1(1,1) is not in the set (we put a constraint that x&y has to be >= 0). it's not close under scalar multiplication
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