Bildet R2{\displaystyle R_{2}} mit den angegebenen Operationen einen Vektorraum über R{\displaystyle R}? (x1,x2)+(y1,y2)=(x1+y2,x2+y1),λ(x1,x,2)=(λx1,λx2){\displaystyle (x_{1},x_{2})+(y_{1},y_{2})=(x_{1}+y_{2},x_{2}+y_{1}),\lambda (x1,x,2)=(\lambda x1,\lambda x2)}
((a1a2)+(b1b2))+(c1c2)=(a1+b2a2+b1)+(c1c2)=(a1+b2+c2a2+b1+c1){\displaystyle ({\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}})+{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{2}\\a_{2}+b_{1}\end{pmatrix}}+{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{2}+c_{2}\\a_{2}+b_{1}+c_{1}\end{pmatrix}}}
(a1a2)+((b1b2)+(c1c2))=(a1a2)+(b1+c2b2+c1)=(a1+b2+c1a2+b1+c2){\displaystyle {\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+({\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}+{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}})={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+{\begin{pmatrix}b_{1}+c_{2}\\b_{2}+c_{1}\end{pmatrix}}={\begin{pmatrix}a_{1}+b_{2}+c_{1}\\a_{2}+b_{1}+c_{2}\end{pmatrix}}}
(R2,+){\displaystyle (\mathbb {R} ^{2},+)} ist nicht assoziativ, daher auch kein Vektorraum.