Is $\text{Im}(T) + \text{Ker }(T)$ the same as $\text{Im}(T)$ union $\text{Ker }(T)$ If I know $\text{Im}(T)$ and $\text{Ker }(T)$, is $\text{Im}(T)+\text{Ker }(T)$ the union of the two vector space?
If not, how do I find the addition of the two vector space. It is best if examples can be given. Thanks.
 A: No, ${\rm Im}(T)+{\rm Ker}(T)$ is not the same as ${\rm Im}(T)\cup {\rm Ker}(T)$. The former is defined as
$$
{\rm Im}(T)+{\rm Ker}(T) = \{x+y: x \in {\rm Im}(T), y \in {\rm Ker}(T) \}
$$
To visualize this, imagine ${\rm Im}(T)$ is the $x$-axis in $\mathbb{R}^3$, and ${\rm Ker}(T)$ is the $yz$-plane. Then ${\rm Im}(T)+{\rm Ker}(T)$ would be the entire $\mathbb{R}^3$, since any vector in $\mathbb{R}^3$ can be written as a sum of a vector on $x$-axis and a vector on $yz$-plane. On the other hand, the union of the two would not contain any point outside the $x$-axis and the $yz$-plane.
A: For an example, let $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be defined such that
$$T(x,y)=T(x,0)$$
so $T$ is essentially the projection of a vector in $\mathbb{R}^2$ on the x axis. Thus with this we have that
$${\rm Im}(T)=\{(a,0):a\in\mathbb{R}\}$$
and
$${\rm Ker}(T)=\{(0,b):b\in\mathbb{R}\}$$
Thus we have that ${\rm Im}(T)\cup {\rm Ker}(T)$ is just set contains the $x$ axis and $y$ axis, but notice that ${\rm Im}(T)+{\rm Ker}(T)=\{(a,b):a,b\in\mathbb{R}\}=\mathbb{R}^{2}$
