Can someone help me show these two linear algebra problems?

Question

I really don't understand this stuff :(
But I'll actually choose a best answer!

Two linear algebra problems:

1. Let A and B be any two matrices such that the product AB is defined. Show that Ker(B) is a subset of Ker(AB).

2. Let A and B be any two matrices such that the product AB is defined. Suppose that Ker(A) = 0. Show that in this case Ker(B) = Ker(AB).

Thanks in advance! :)

David · Accepted Answer

1. Ker(B) is the set of all vectors v, such that B(v) = 0, the 0-vector.

suppose v is in Ker(B), so that B(v) = 0.

then (AB)(v) = A(B(v)) = A(0) = 0, so v is in Ker(AB), thus Ker(B) is a subset of Ker(AB).

we already know from part (1) that Ker(B) is always a subset of Ker(AB).

suppose now, that the ONLY vector in Ker(A) is the 0-vector.

if v is in ker(AB), then (AB)(v) = A(B(v)) = 0

thus B(v) is in ker(A), and must be the 0-vector, so B(v) = 0.

hence v is in ker(B), so in this case, ker(AB) is a subset of ker(B), as well.

the only way this can happen is if ker(AB) = ker(B).

moton · Answer

i do no longer understand the small print, yet any matrix A with determinant no longer equivalent to 0 is an invertible (nonsigular) matrix. If A is nonsingular, then that's invertible. besides, the inverse of A, denoted as A^(-a million) is unique. So the equation MX = V has a special answer. it relatively is (ii) implies (i). i do no longer understand approximately (i) implies (ii). wish this helps

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Can someone help me show these two linear algebra problems?

2 Answers