[Math] Find bases for N(T) and R(T) and verify the Dimension Theorem

linear algebra

We have
$$
T\pmatrix{a11&a12&a13\\a21&a22&a23\\a31&a32&a33}= \pmatrix{a11+a12+a13&a21+a22+a23\\a21+a22+a23&0}.
$$

I know what the Dimension Theorem is and by looking at the equation I can tell that $a11+a12=-a13$ and so on shows that the null space contains more than just the zero vector, however I struggle in figuring out how to express the bases of $N(T)$ and $R(T)$.

Best Answer

You can just try some small numbers, easiest to play with $1$ and $0$.

For $N(T)$, as you started you get these $3$ equations ($2$ of them are the same) as $-a_{11}-a_{12} =a_{13}$, so that $a_{11}$, $a_{12}$ and $a_{21}$, $a_{22}$ can be anything, only $a_{13}$ and $a_{23}$ are bound to them. Also, $a_{31},\ a_{32},\ a_{33}$ can be anything.

So, consider these elements of $N(T)$:

  • $a_{11}=1$, rest is $0$ ($\implies a_{13}=-1$)
  • $a_{12}=1$, rest is $0$ ($\implies a_{13}=-1$)
  • $a_{21}=1$, rest is $0$ ($\implies a_{23}=-1$)
  • and so on...
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