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 \(\displaystyle{M}_{{{2}{x}{4}}}\).

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 \(\displaystyle{M}_{{{2}{x}{4}}}\).