A subspace is a subset H of a vector space V with the following 3 characteristics:
1. the zero vector of V is in H
2. the subspace gets sealed as more is added.
3. in the case of scalar multiplication, the subspace is closed.
1. Let A = 0. Then for any matrix F, FA = 0. Thus, . So H contains the zero vector.
2. Let . Consider due to the distributive property of matrices. Since .
Thus, , so H is closed under addition.
3. Let . Consider cA. F(cA) = c(FA) according to scalar properties. Since , c(FA) =c(0) =0. Thus, . So H is closed under scalar multiplication.
Thus, H fulfills all the requirements of the definition of a subspace of .
Find the vector and parametric equations for the line segment connecting P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)