Question

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)

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

2021-02-12
A subspace of a vector space V is a subset H of V with 3 properties:
1. the zero vector of V is in H
2. the subspace is closed under addition
3. the subspace is closed under scalar multiplication.
1. Let A = 0. Then for any matrix F, FA = 0. Thus, $$\displaystyle{A}\in{H}$$. So H contains the zero vector.
2. Let $$\displaystyle{A}_{{1}},{A}_{{2}}\in{H}$$. Consider $$\displaystyle{A}_{{1}}+{A}_{{2}}.{F}{\left({A}_{{1}}+{A}_{{2}}\right)}={F}{A}_{{1}}{4}+{F}{A}_{{2}}$$ due to the distributive property of matrices. Since $$\displaystyle{A}_{{1}},{A}_{{2}}\in{H},{F}{A}_{{1}}+{F}{A}_{{2}}={0}+{0}={0}$$.
Thus, $$\displaystyle{A}_{{1}}+{A}_{{2}}\in{H}$$, so H is closed under addition.
3. Let $$\displaystyle{c}\in{R},{A}\in{H}$$. Consider cA. F(cA) = c(FA) according to scalar properties. Since $$\displaystyle{A}\in{H}$$, c(FA) =c(0) =0. Thus, $$\displaystyle{c}{A}\in{H}$$. So H is closed under scalar multiplication.
Thus, H fulfills all the requirements of the definition of a subspace of $$\displaystyle{M}_{{{2}{x}{4}}}$$.