- #1
gruba
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Homework Statement
Let [itex]U[/itex] is the set of all commuting matrices with matrix [itex]A= \begin{bmatrix}
2 & 0 & 1 \\
0 & 1 & 1 \\
3 & 0 & 4 \\
\end{bmatrix}[/itex]. Prove that [itex]U[/itex] is the subspace of [itex]\mathbb{M_{3\times 3}}[/itex] (space of matrices [itex]3\times 3[/itex]). Check if it contains [itex]span\{I,A,A^2,...\}[/itex]. Find the dimension and a basis for [itex]U[/itex] and [itex]span\{I,A,A^2,...\}[/itex].
Homework Equations
-Commuting matrices
-Subspaces
-Vector space span
Basis and dimension
The Attempt at a Solution
[itex]U[/itex] can be defined as [itex]U=\{B\in\mathbb{M_{3\times 3}}: AB=BA\}[/itex].
Letting [itex]B=\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}[/itex] and solving the equation [itex]AB=BA[/itex] gives [itex]B=\begin{bmatrix}
i-\frac{2}{3}g & 0 & \frac{1}{3}g \\
g-3f & i-3g & f \\
g & 0 & i \\
\end{bmatrix}[/itex].
[itex]U[/itex] is a subspace of [itex]\mathbb{M_{3\times3}}[/itex] if [itex]\forall u_1,u_2\in U\Rightarrow u_1+u_2\in U,\forall t\in\mathbb{R}\Rightarrow tu_1\in U[/itex] which is correct.
It is easy to check that if [itex]C\in span\{I,A,A^2,...\}\Rightarrow C\in U[/itex]:
Linear combination for C is
[itex]C=c_0I+c_1A+c_2A^2+...\Rightarrow CA=AC\Rightarrow C\in U[/itex]
[itex]U[/itex] has the dimension [itex]3[/itex] and a basis are column vectors of identity matrix [itex]3\times 3[/itex].
How to find the dimension and a basis for [itex]span\{I,A,A^2,...\}[/itex]?