- #1
gruba
- 206
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Homework Statement
Let [itex]\mathcal{A}: \mathbb{R^3}\rightarrow \mathbb{R^3}[/itex] is a linear operator defined as [itex]\mathcal{A}(x_1,x_2,x_3)=(x_1+x_2-x_3, x_2+7x_3, -x_3)[/itex]
Prove that [itex]\mathcal{A}[/itex] is invertible and find matrix of [itex]\mathcal{A},A^{-1}[/itex] in terms of canonical basis of [itex]\mathbb{R^3}[/itex].
Homework Equations
-Linear transformations
-Jordan decomposition
The Attempt at a Solution
[/B]
Linear mapping can be written as
[tex]\mathcal{A}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix}
=
\begin{bmatrix}
x_1+x_2-x_3 \\
x_2+7x_3 \\
-x_3 \\
\end{bmatrix}
[/tex]
Matrix of linear operator [itex]\mathcal{A}[/itex] is [tex]T=
\begin{bmatrix}
1 & 1 & -1 \\
0 & 1 & 7 \\
0 & 0 & -1 \\
\end{bmatrix}
[/tex]
Using Jordan decomposition method, it is possible to find [itex]T,T^{-1}[/itex] in terms of canonical basis.
How to strictly prove that [itex]\mathcal{A}[/itex] is invertible (without matrix computation)?