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If i know Im(T) and Ker(T), is Im(T) + 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.

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    $\begingroup$ In many cases the union of vector spaces in not a vector space while sum of vector spaces is. A sum of vector spaces can be thought as the vector space consisting of sum of all elements of each vector space. Since vector spaces can be generalize by span of a basis of vectors you could find the what the sum of vector spaces by seeing what span of union of both basis are $\endgroup$ – Kamster Sep 22 '14 at 6:32
  • $\begingroup$ also Im a little confused whether if $Im(T)+Ker(T)$ is well defined because we could have $Im(T)$ and $Ker(T)$ be different dimension and I dont know how to add vectors such as $(a,b)+(a',b',c')$. If they are the same dimension this statement makes sense $\endgroup$ – Kamster Sep 22 '14 at 7:26
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No, $Im(T)+Ker(T)$ is not the same as $Im(T)\cup Ker(T)$. The former is defined as

$$ Im(T)+Ker(T) = \{x+y: x \in Im(T), y \in Ker(T) \} $$

To visualize this, imagine $Im(T)$ is the $x$-axis in $\mathbb{R}^3$, and $Ker(T)$ is the $yz$-plane. Then $Im(T)+Ker(T)$ 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.

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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 $$Im(T)=\{(a,0):a\in\mathbb{R}\}$$ and $$Ker(T)=\{(0,b):b\in\mathbb{R}\}$$ Thus we have that $Im(T)\cup Ker(T)$ is just set contains the $x$ axis and $y$ axis, but notice that $Im(T)+Ker(T)=\{(a,b):a,b\in\mathbb{R}\}=\mathbb{R}^{2}$

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