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Matrix kernel
Task number: 2510
Let A be a matrix of size m×n over a field K. Show that Ker(A) forms a vector subspace in the arithmetic vector space Kn.
Hint
Use the definition of a matrix kernel and show that the set is closed under summation and product with any element of the field K.
Resolution
Let x,x′∈Ker(A), then from the definition Ax=Ax′=0, then also A(x+x′)=Ax+Ax′=0+0=0, and so x+x′∈Ker(A).
Similarly let x∈Ker(A) and a∈K then we compute A(ax)=a(Ax)=a0=0, and so ax∈Ker(A).
Note that although there are three different multiplications in the first equation, it is possible to change the order in which the product with scalar will be done.
