The Dimension of The Null Space and Range Examples 1
Recall from The Dimension of The Null Space and Range page that if $T$ is a linear map from $V \to W$ and $V$ is finite-dimensional then we have the following formula relating the dimension of $V$ to the dimension of the the null space of $T$ and the dimension of the range of $T$:
(1)We will now look at some examples applying this formula.
Example 1
Let $U$, $V$, and $W$ be finite-dimensional vector spaces, and suppose that $S \in \mathcal L (V, W)$ and $T \in \mathcal L (U, V)$. Let $ST \in \mathcal L (U, W)$. Show that $\mathrm{dim} (\mathrm{null} (ST)) ≤ \mathrm{dim} ( \mathrm{null} (S)) + \mathrm{dim} ( \mathrm{null} (T))$.
Using the dimension formula above we have that:
(2)Since $\mathrm{dim} (U) = \mathrm{dim} (\mathrm{null} (T)) + \mathrm{dim} (\mathrm{range} (T))$ we can substitute this into the equation above to get that:
(3)Now notice that $\mathrm{dim} (\mathrm{range}(T)) ≤ \mathrm{dim} (V)$ and so:
(4)Since $\mathrm{dim} (V) = \mathrm{dim} (\mathrm{null} (S)) + \mathrm{dim} (\mathrm{range} (S))$, we can substitute this into the equation above to get:
(5)Note that $\mathrm{dim} (\mathrm{range} (S)) - \mathrm{dim} (\mathrm{range} (ST)) ≤ 0$ though since $S$ maps elements $v \in V$ to elements $S(v) \in W$, and $ST$ maps elements $T(u) \in V$ to elements $S(T(u)) \in W$ and so $\mathrm{range} S \subseteq \mathrm{range} (ST)$ which implies that $\mathrm{dim} ( \mathrm{range} (S)) ≤ \mathrm{dim} ( \mathrm{range} (ST))$, so:
(6)Example 2
Let $V$ be a finite-dimensional vector space. Suppose that $V = \mathrm{null}(T) + \mathrm{range}(T)$. Prove that $\mathrm{null} (T) \cap \mathrm{range} (T) = \{ 0 \}$.
If we apply the dimension formula given above, we have that:
(7)Furthermore, from The Dimension of a Sum of Subspaces we have that:
(8)Since $V = \mathrm{null} (T) + \mathrm{range} (T)$, this implies that the two equations above are equal, and so:
(9)The equation above implies that $\mathrm{dim} (\mathrm{null} (T) \cap \mathrm{range} (T)) = 0$, so $\mathrm{null} (T) \cap \mathrm{range} (T) = \{ 0 \}$.