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)

Maiclubk 2021-02-11 Answered
Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M2×4 with the property that FA = 0 (the zero matrix in M3×4). Determine if H is a subspace of M2×4
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Expert Answer

yagombyeR
Answered 2021-02-12 Author has 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, AH. So H contains the zero vector. 
2. Let A1,A2H. Consider A1+A2.F(A1+A2)=FA14+FA2 due to the distributive property of matrices. Since A1,A2H,FA1+FA2=0+0=0
Thus, A1+A2H, so H is closed under addition. 
3. Let cR,AH. Consider cA. F(cA) = c(FA) according to scalar properties. Since AH, c(FA) =c(0) =0. Thus, cAH. So H is closed under scalar multiplication. 
Thus, H fulfills all the requirements of the definition of a subspace of M2x4.

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