Task list filter?
Choose required ranks and required tasks. The table of contents will list only tasks having one of the required ranks in corresponding rankings and at least one of the required tags (overall). If you wish to filter only according to some rankings or tags, leave the other groups empty.
Task rankings
Difficulty level
Task tags
Task type
«
«
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.