Exercise 1915Linear algebra › Vector spaces and subspacesL1Added 3 Nov 2026

Intersection of two kernels

#intersection#kernel#dimension

Statement

Let $f,g:\mathbb{R}^3\to\mathbb{R}$ be the linear forms $f(x,y,z)=x+y+z$ and $g(x,y,z)=x-y$ . Compute $\dim(\ker f\cap\ker g)$ .

Hint

The intersection is the kernel of the map $(f,g):\mathbb{R}^3\to\mathbb{R}^2$ .

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Map. $(f,g)(x,y,z)=(x+y+z,x-y)$ .

Matrix.
$\begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \end{pmatrix}.$
The two rows are not proportional, so the rank is $2$ .

Dimension. $\dim(\ker f\cap\ker g)=3-2=1$ .

Explicit. Solve $x+y+z=0$ and $x=y$ : $2x+z=0$ , so $z=-2x$ . Kernel is spanned by $(1,1,-2)$ .

Conclusion. $\ker f\cap\ker g$ is the line spanned by $(1,1,-2)$ .