Largest/smallest dimensions possible?

Question

If $T : \mathbb R^5 \to \mathbb R^2$ is a linear transformation, what is the smallest dimension that $\ker(T)$ can have? If $L : \mathbb R^2 \to \mathbb R^4$ is a linear transformation, what is the largest dimension that $\operatorname{Range}(L)$ can have? What is the smallest dimension that $\ker(LT)$ can have, where $LT : \mathbb R^5 \to \mathbb R^4$ is the composition of $L$ and $T$, $LT(v) = L(T(v))$? What is the largest dimension that $\operatorname{Range}(LT)$ can have?

I assume I am using the theorem that $\dim(\operatorname{Range}(T)) + \dim(\ker(T)) = \dim(V)$, right? (Assuming Let $T:V→W$)

How do I go about solving for the missing variables?

Answer 1

Hint:

As a first step notice that, for a linear transformation $L:\mathbb{R}^n \to \mathbb{R}^m$, the largest dimension for the range is $m$. This, with the mentioned rank-nullity theorem, solve immediately your first question.

And, with a bit more of reasoning, also the other questions.

Answer 2

Every linear transformation is equivalent to some matrix. For the first case the matrix is $M_{5\times 2}$ whose rank is at most $2$ and according to rank-nullity theorem the kernel space dimension is at least $3$. For the 2nd case similarly the range space dimension is at most $2$. You can easily solve the other cases.