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Let $T$ be the linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Find an example of such $T$ where $\ker T = \text{im } T$.

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    $\begingroup$ My bad about the incorrect example. Rank-Nullity tells you $\dim(\ker T)+\dim(\text{Im } T)=2 \implies \dim(\ker T)=\dim(\text{Im } T)=1$ $\endgroup$
    – Shuri2060
    Jul 30, 2017 at 14:49
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    $\begingroup$ Possible duplicate of Linear Transformation with kernel equal to image $\endgroup$
    – user228113
    Jul 30, 2017 at 15:04
  • $\begingroup$ In addition, I believe all such examples are projections (and this also generalizes to higher even dimensions) if you consider the problem geometrically. $\endgroup$
    – Shuri2060
    Jul 30, 2017 at 15:10

1 Answer 1

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Consider the linear map defined by $T(1,0) := (0,0)$ and $T(0,1) = (1,0)$. In matrix form,

$$ \begin{bmatrix}0 & 1 \\0 & 0\end{bmatrix}.$$

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