Maiclubk

2021-02-11

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

yagombyeR

Skilled2021-02-12Added 92 answers

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 $A}_{1}+{A}_{2}.F({A}_{1}+{A}_{2})=F{A}_{1}4+F{A}_{2$ 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 $M}_{2x4$.

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