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authorAdrian Kummerlaender2016-09-12 21:32:18 +0200
committerAdrian Kummerlaender2016-09-12 21:32:18 +0200
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Add information on (direct) sums of vector subspaces
-rw-r--r--lineare_algebra.tex14
1 files changed, 14 insertions, 0 deletions
diff --git a/lineare_algebra.tex b/lineare_algebra.tex
index 682dffe..b5a3358 100644
--- a/lineare_algebra.tex
+++ b/lineare_algebra.tex
@@ -188,6 +188,20 @@ Seien $K$ Körper, $V$ $K$-Vektorraum und $U \subseteq V$. Dann ist äquivalent:
$U \subset V$ ist $\phi$-invariant, wenn $\phi(U) \subset U$.
+\subsubsection*{Summe}
+
+$U_1 + U_2 = {u_1+u_2 | u_1 \in U_1, u_2 \in U_2 }$ ist Summe von $U_1$ und $U_2$, kleinster UVR der $U_1 \cup U_2$ enthält.
+
+$dim(U_1 + U_2) = dim(U_1) + dim(U_2) - dim(U_1 \cap U_2)$
+
+\subsubsection*{Direkte Summe}
+
+$U_1 + U_2$ ist direkte Summe, wenn $U_1 \cap U_2 = {0}$.
+
+d.h. gdw. der Schnitt eines UVR mit der Summe aller anderen UVR nur den Nullvektor enthält.
+
+$dim(U_1 \oplus U_2)=dim(U_1)+dim(U_2)$
+
\subsection*{Homomorphismen}
Seien $V, W$ zwei $K$-Vektorräume. $\phi : V \rightarrow W$ ist Vektorraumhomomorphismus respektive $K$-lineare Abbildung, wenn: