If you have a linear transformation T:V→W and the dimension of V is greater than the dimension of W, then the kernel of T has to have a dimension at least as large as the difference.
For example, a transformation $T:\mathbb{R}^7\rightarrow \mathbb{R}^3$ has to have a kernel of dimension 4 or greater.
May I use this?
$\dim \ker(T)+ \dim R(T) = \dim V$
$\dim \ker (T) = \dim V - \dim R(T)$
But $R(T)$ belongs to $W$, so the dimension of $R(T)$ has to be equal or less than dimension of $W$ (How can I prove this?). So,
$\dim \ker (T) \geq \dim V - \dim W$
And $\dim V > \dim W$, so $\dim V - \dim W$ is positive.
Is this right?
May I use this property to say when V is infinite dimensional and W has a finite dimension, then the kernel has to be infinite dimensional?
Thank you very much for your help.