Find the range and kernel of $T$.
a)
$T(v_{1}, v_{2}) = (v_{2}, v_{1})$
For this one, I think the range is the span of bases $(0,1), (1,0)$. Since $v_{1}$ and $v_{2}$ are switched. As for its kernel, it should be the span of basis $(0,0)$, but I'm not quite sure if the zero vector can be a basis.
b)
$T(v_{1}, v_{2}, v_{3}) = (v_{1}, v_{2})$
Range: span of bases $(1,0), (0,1)$. Kernel is the span of basis $(0,0)$.
c)
$T(v_{1}, v_{2}) = (0, 0)$
Range & kernel both the span of $(0,0)$.
d)
$T(v_{1}, v_{2}) = (v_{1}, v_{1})$
Range: span of basis $(1,0)$. Kernel is the line $v_{1} = 0$ since we're in $\mathbb{R}^{2}$
Are my answers correct?
Best Answer
You made some mistakes:
(a): Range is all the space, while the kernel is the zero-vector along.
(b): The range is the whole of $\mathbb R^2,$ while the kernel, a subspace of $\mathbb R^3,$ is the subspace of $\mathbb R^3$ generated by $(0,0,1).$
(c): The range is spanned by $(0,0)$ indeed, but the kernel is not: it is the whole $\mathbb R^2.$
(d): The range is spanned by $(1,1).$ And the kernel is spanned by $(0,1)$.
Hope this helps.