Lineare Algebra 1/Gemischte Satzabfrage/30/Aufgabe/Lösung

${\displaystyle {}K}$

${\displaystyle {}V}$

${\displaystyle {}K}$

${\displaystyle {}n}$

${\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}$

${\displaystyle {}{\mathfrak {u}}=u_{1},\ldots ,u_{n}}$

${\displaystyle {}V}$

${\displaystyle {}v_{j}=\sum _{i=1}^{n}c_{ij}u_{i}\,}$

mit den Koeffizienten ${\displaystyle {}c_{ij}\in K}$, die wir zur ${\displaystyle {}n\times n}$-Matrix

${\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}={\left(c_{ij}\right)}_{ij}\,}$

zusammenfassen. Dann hat ein Vektor ${\displaystyle {}w}$, der bezüglich der Basis ${\displaystyle {}{\mathfrak {v}}}$ die Koordinaten ${\displaystyle {}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}}$ besitzt, bezüglich der Basis ${\displaystyle {}{\mathfrak {u}}}$ die Koordinaten

${\displaystyle {}{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}=M_{\mathfrak {u}}^{\mathfrak {v}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}c_{11}&c_{12}&\ldots &c_{1n}\\c_{21}&c_{22}&\ldots &c_{2n}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\ldots &c_{nn}\end{pmatrix}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\,.}$

${\displaystyle {}\lambda \in K}$

${\displaystyle {}\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}\leq \mu _{\lambda }(\varphi )\,.}$