is the following set a subspace of v3 {x,y,z|x,y=0}

there is a test for subspace. make sure it has a zero element (0,0,0) and is closed under multiplication (we dont have to test all the axioms of a vector space)

so it has the zero element, set z = 0. and it is closed under multiplication c(0,0,z) = (0,0,cz) and (0,0,cz) is an element of { x, y , z | x , y = 0 }

Oh i left out one more thing. we have to make sure addition is closed. so if (0,0,z1) + (0,0,z2) show that it is an element of {x,y,z|x,y=0} which is easy, because the sum is (0,0,z1+z2) and thats an element of {x,y,z|x,y=0}. Here are conditions for linear subspace. Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following 3 conditions: 1. The zero vector, 0, is in W. 2. If u and v are elements of W, then any linear combination of u and v is an element of W; 3. If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

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