# Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M_(2×4) with the property that FA = 0 (the zero matrix in M_(3×4)). Determine if H is a subspace of M_(2×4)

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in ${M}_{2×4}$ with the property that FA = 0 (the zero matrix in ${M}_{3×4}\right)$. Determine if H is a subspace of ${M}_{2×4}$
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yagombyeR

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, $A\in H$. So H contains the zero vector.
2. Let ${A}_{1},{A}_{2}\in H$. Consider ${A}_{1}+{A}_{2}.F\left({A}_{1}+{A}_{2}\right)=F{A}_{1}4+F{A}_{2}$ due to the distributive property of matrices. Since ${A}_{1},{A}_{2}\in H,F{A}_{1}+F{A}_{2}=0+0=0$
Thus, ${A}_{1}+{A}_{2}\in H$, so H is closed under addition.
3. Let $c\in R,A\in H$. Consider cA. F(cA) = c(FA) according to scalar properties. Since $A\in H$, c(FA) =c(0) =0. Thus, $cA\in H$. So H is closed under scalar multiplication.
Thus, H fulfills all the requirements of the definition of a subspace of ${M}_{2x4}$.