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For the first part I put the matrix into echleon form :


And then:

$r(A)=2\implies \dim(A)=4-2=2$

^^ Is my presentation correct?

The second part is the biggest problem. I don't know how to do. Please help.

Also if anyone knows a good book or a good website to master this chapter so that I can answer many questions easily, let me know. Would be very helpful to me.

  • $\begingroup$ For the second part: If $M:\mathbb{R}^4 \to \mathbb{R}^4$ is non-singular, what does this imply about the kernel of $M$ and thus the range of $M$? $\endgroup$ – Krijn Oct 20 '14 at 10:59
  • $\begingroup$ If $M$ is non-singular, how do the dimensions of the domain of $M$ and the range of $M$ relate? $\endgroup$ – Ruvi Lecamwasam Oct 20 '14 at 11:00
  • $\begingroup$ @AndrewLedesma I don't know :( I know that M has an inverse. What does domain of a matrix mean? $\endgroup$ – user181306 Oct 20 '14 at 11:01
  • $\begingroup$ The domain of $M$ is the space that $M$ acts on, i.e the domain of $A$ above is $\mathbb{R}^4$. If $\{v_i\}$ in the domain of $M$ are linearly independent, what do you know about $\{Mv_i\}$? (consider applying $M^{-1}$ to $\sum_n Mc_nv_n$ for $c_n$ arbitrary constants.) $\endgroup$ – Ruvi Lecamwasam Oct 20 '14 at 11:03
  • $\begingroup$ @Krijn what about its null space? $\endgroup$ – user181306 Oct 20 '14 at 11:03

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