Task number: 2510
Let \(\mathbf A\) be a matrix of size \(m \times n\) over a field \(\mathbb K\). Show that \(Ker(\mathbf A)\) forms a vector subspace in the arithmetic vector space \(\mathbb K^n\).
Use the definition of a matrix kernel and show that the set is closed under summation and product with any element of the field \(\mathbb K\).
Let \(\mathbf x,\mathbf x' \in Ker(\mathbf A)\), then from the definition \(\mathbf A\mathbf x=\mathbf A\mathbf x'=\mathbf 0\), then also \(\mathbf A(\mathbf x+\mathbf x')=\mathbf A\mathbf x+\mathbf A\mathbf x'=\mathbf 0+\mathbf 0=\mathbf 0\), and so \(\mathbf x+\mathbf x'\in Ker(\mathbf A)\).
Similarly let \(\mathbf x\in Ker(\mathbf A)\) and \(a\in \mathbb K\) then we compute \(\mathbf A(a\mathbf x)=a(\mathbf A\mathbf x)=a\mathbf 0=\mathbf 0\), and so \(a\mathbf x\in Ker(\mathbf A)\).
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.