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For fixed positive integers m and​ n, the set Upper M Subscript m times n of all mtimesn matrices is a vector​ space, under the usual operations of addition of matrices and multiplication by real scalars. Let F be a fixed 3times2 ​matrix, and let H be the set of all matrices A in Upper M Subscript 2 times 4 with the property that FAequals0 ​(the zero matrix in Upper M Subscript 3 times 4​). Determine if H is a subspace of Upper M Subscript 2 times 4.

Respuesta :

Answer:

Yes, H is a subspace

Step-by-step explanation:

Recall that given a vector space V, a subset W of V is a subspace if and only if

- the 0 of V is in W

- given a, b in W, then a+b is in W

- given a real scalar r and a in W, then ra is in W.

In order to see if H is a subspace of [tex]M^{2\times 4}[/tex] we must check the three properties.

- It is clear that the matrix 0 in  [tex]M^{2\times 4}[/tex] is in H since [tex]F0 = 0[/tex] (where the right hand 0 is the 0 vector in [tex]M^{3\times 4}[/tex].

- Let A,B in H. We want to check that A+B is in H. Since A,B in H we have that FA=0 and FB=0. We have that

[tex]F(A+B) = FA+FB = 0 +0 =0[/tex]

Then A+B is in H.

- given a real number r, and a matrix A in H, we want to check if rA is in H. Then

[tex] F(rA) = rFA = r0 = 0[/tex]

which shows that rA is in H.

Hence, H is a subspace of [tex] M^{2\times 4}[/tex]

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