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Task149

Let \(e_1, e_2, e_3\) be the standard basis of \(\mathbb{R}^3\):
\begin{equation*}
e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad
e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad
e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.
\end{equation*}
Make an example of an invertible \(3 \times 3\) matrix \(S\).
Write your matrix as a matrix of column vectors.
\begin{equation*}
S = \begin{pmatrix} | & | & | \\ v_1 & v_2 & v_3 \\
| & | & | \end{pmatrix}
\end{equation*}
How do you know that the set \(\{ Se_1, Se_2, Se_3 \}\) is a basis
for \(\mathbb{R}^3\)?

What is the connection between \(Se_1\), \(Se_2\), \(Se_3\),
\(S^{-1}v_1\), \(S^{-1}v_2\), \(S^{-1}v_3\) and the original
vectors \(e_1, e_2, e_3, v_1, v_2, v_3\)?

Finally, how do we use this to understand the way that the decomposition
\(A = S\Lambda S^{-1}\) works?