## Matrix kernel

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 $$\mathbb K^n$$.

• #### 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 $$\mathbb K$$.

• #### Resolution

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.  