Help finding a linear transformation

Question

I'm a little bit lost here; I need to find a linear transformation $T:\mathbb{R}^3\to\mathbb{R}^3$ such that $\ker(T)\cap\text{Im}(T)\neq\left\lbrace 0\right\rbrace$.

I have been looking for it all day, but any linear transformation I choose, I get $\ker(T)\cap\text{Im}(T)=\left\lbrace 0\right\rbrace$. Any help or idea will be really helpful.

Have you tried in $\mathbb R^2$? – J. W. Tanner May 12 '21 at 02:13 — J. W. Tanner, May 12 '21 at 02:13

score 1 · Accepted Answer · answered May 12 '21 at 02:16

1

Say $T$ maps $(1,0,0)$ to $(0,0,0)$ and $(0,1,0)$ to $(1,0,0)$ and $(0,0,1)$ to $(0,0,1)$.

Then what do you get for $\ker(T)\cap\text{Im}(T)$?

answered May 12 '21 at 02:16

J. W. Tanner

60,406

It can't be, right? I mean, a linear transformation carry linear independent sets to linear independent sets – Pech May 12 '21 at 02:27
No, linear independence does not need to be preserved; for example, consider the linear transformation that maps all the vectors to $\mathbf 0$ – J. W. Tanner May 12 '21 at 02:34
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You'r right! and also te mapping in this case is linear independent, and I get that $\left\lbrace(1,0,0)\right\rbrace\in\ker(T)\cap\text{Im}(T)$ thank you so much!!! – Pech May 12 '21 at 02:55

Help finding a linear transformation

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