Review of Linear Algebra Basics
1.1. Review of Linear Algebra Basics#
To use quantum mechanics, you should be familiar with linear algebra operations. The following exercises help you to review some basic operations with vectors and matrices.
Exercise 1
Calculate the vector product \(\mathbf{c}=\mathbf{a}\times\mathbf{b}\) with
\[\begin{split}
\begin{aligned}
\mathbf{a} = \left(\begin{matrix}
2 \\ 6 \\ 4
\end{matrix}\right)
\quad
\mathbf{b} = \left(\begin{matrix}
5 \\ 1 \\ 7
\end{matrix}\right)
\end{aligned}
\end{split}\]
and, for the same \(\mathbf{a},\mathbf{b}\), the scalar product
\[
\begin{aligned}
d = \mathbf{a}\cdot \mathbf{b}
\end{aligned}
\]
Exercise 2
Evaluate the matrix product \(\mathbf{C} = \mathbf{A}\mathbf{B}\).
\[\begin{split}
\mathbf{A} =\begin{aligned}
\left(\begin{matrix}
6 & 8 &2 \\ 9 & 1 & 5 \\ 7 & 4 & 3
\end{matrix}\right),\quad\mathbf{B}=
\left(\begin{matrix}
9 & 6 & 7 \\ 5 & 4 & 4 \\ 3 & 2 & 8
\end{matrix}\right).
\end{aligned}
\end{split}\]
Exercise 3
Evaluate the determinant for the matrix \(\mathbf{A}\).
\[\begin{split}
\mathbf{A}=\begin{aligned}
\left(\begin{matrix}
1 & 1 & 2 \\ 1 & 0 & 2 \\ 2 & 2 & 3
\end{matrix}\right)
\end{aligned}
\end{split}\]
Exercise 4
Does the exponent of an operator always satisfy the relation \(e^{\hat{A}+\hat{B}} = e^{\hat{A}}e^{\hat{B}}\) ? Start from the definition of the matrix exponential.