Counter examples for Linear transformation from $:V\to V$

Question

Using matrices , find examples as called for below

(a) Find a linear transformation $T:V\to V$ such that $\ker T \neq 0$ but that $T$ is not surjective

(b) Find a linear transformation $T:V\to V$ such that $T\ne I_V$ , but $T^2=T$

(c) Find a linear transformation $T:V\to V$ such that $T\ne 0$ , but $T^2=0_V$

(d) Find a linear transformation $T:V\to V$ such that $T^k\ne 0_v$ but $T^{k+1}=0_V$

(e) Find a linear transformation $T:V\to V$ such that $T^4=I_v$, but $T^k \ne I_v$ if $0<k<4$

for (a) is $T(x,y)=(0,y)$ is correct?

(b) $T(x,y)=(x,0)$ is correct?

(c) $T(x,y)=(y,0)$ correct?

remaining i dont know can you some one please?

Answer 1

If you can choose any $V$ you want then for $(d)$ you can take $V=\mathbb{R}^{k+1}$ and $T(x_1,x_2,...,x_{k+1})=(x_2,x_3,...,x_{k+1},0)$.

For $(e)$ take $V=\mathbb{R^4}$ and $T(x_1,x_2,x_3,x_4)=(x_4,x_1,x_2,x_3)$.

Answer 2

Assuming $V$ can be anything, your first three answers will be correct if you specify e.g. $V = \mathbb{C}^2$ or something like that.

For (d) use a similar idea as for (c), just in $k+1$-dimensions: on $V = \mathbb{C}^{k+1}$ define $$T(x_1, \ldots, x_{k+1}) = (0, x_1, \ldots, x_k)$$

For (e), you can use that the multiplicative order of $i \in \mathbb{C}$ is $4$ so on $V = \mathbb{C}$ define $$T(x) = i\cdot x$$