Definition 4.1.1. Matrices.
A matrix \(A=\{a_{ij}\}\) is any rectangular array of numbers. The size of \(A\) is written as \(m \times n\text{,}\) where \(m\) is the number of rows and \(n\) is the number of columns.
Section 4.1 Introduction to linear systems
Theorem 4.1.2. Properties of matrices.
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Equality: \(A=B\) if and oly if \(A\) and \(B\) are the same size and \(a_{ij}=b_{ij}\) for all \((i,j)\text{.}\)
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Addition: If \(A\) and \(B\) are the same size, then \(A+B=(a+b)_{ij}\text{.}\)
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Scalar multiplication: If \(A\) is an \(m \times n\) matrix and \(k \in \mathbb{R}\text{,}\) then \(kA=k(a_{ij})\text{.}\)
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Matrix multiplication: If \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, then \(AB\) exists and is an \(m \times p\) matrix.
Example 4.1.3.
Let \(A=\begin{bmatrix} 1&2&3 \\ 4&5&6 \end{bmatrix}\) and \(B=\begin{bmatrix} 1&0 \\ -1&1 \\ 0&1 \end{bmatrix}\text{.}\)
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Find \(AB\text{.}\)
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Find \(BA\text{.}\)
Definition 4.1.4. Matrix transpose.
If \(A=\{a_{ij}\}\) is an \(m \times n\) matrix, its transpose \(A^T=\{a_{ji}\}\) is an \(n \times m\) matrix.
Definition 4.1.5. Square matrices.
A matrix \(A\) is called square if the number of its rows equals the number of its columns, i.e. that \(A\) is an \(n \times n\) matrix.
Definition 4.1.6. Identity matrix.
An identity matrix \(I_n\) is an \(n \times n\) matrix with entries of \(1\) along the main diagonal (top left to bottom right) and \(0\) everywhere else.
Definition 4.1.7. Trace.
The trace of an \(n \times n\) matrix \(A\) is the sum of all the entries on its main diagonal, i.e.
\begin{equation*}
\mathrm{tr}(A) = \displaystyle\sum_{i=1}^n a_{ii}.
\end{equation*}
Definition 4.1.8. Determinant of \(2 \times 2\) matrices.
Let \(A=\begin{bmatrix}a&b \\ c&d\end{bmatrix}\text{.}\) The determinant of \(A\) is
\begin{equation*}
\mathrm{det}(A)=ad-bc.
\end{equation*}
Definition 4.1.9. Invertibility.
Let \(A\) be an \(n \times n\) matrix. If \(\mathrm{det}(A) \neq 0\text{,}\) then \(A\) is called invertible (also called nonsingular), which implies the rows of \(A\) are linearly independent. If \(\mathrm{det}(A)=0\text{,}\) then \(A\) is called noninvertible (also called singular), which implies the rows of \(A\) are linearly dependent so that at least one row can be written as a linear combination of the other rows.
Definition 4.1.10. Matrix inverse.
Let \(A\) be an \(n \times n\) matrix with \(\mathrm{det}(A) \neq 0\text{.}\) Then there exists a unique \(n \times n\) matrix \(A^{-1}\) such that \(A^{-1}A=AA^{-1}=I_n\text{.}\) The matrix \(A^{-1}\) is the multiplicative inverse of \(A\text{.}\)
Theorem 4.1.11. Inverse of \(2 \times 2\) matrices.
If \(A=\begin{bmatrix} a&b\\c&d\end{bmatrix}\) with \(\mathrm{det}(A)=ad-bc\neq 0\text{,}\) then
\begin{equation*}
A^{-1} = \dfrac{1}{\mathrm{det}(A)} \begin{bmatrix} d&-b \\ -c& a\end{bmatrix}.
\end{equation*}
Example 4.1.12.
Consider the linear system
\begin{equation*}
\left\{ \begin{array}{ll} ax+by=s \\ cx+dy=t. \end{array}\right.
\end{equation*}
This sytem can be written in as a vector equation of the form \(A\vec{x}=\vec{b}\text{.}\)
Theorem 4.1.13. Trichotomy for linear systems.
There are three possible outcomes for the solution of the vector equation \(A\vec{x}=\vec{b}\text{.}\) The three possibilities are revealed by understanding a \(2 \times 2\) system as a pair of lines and solutions as intersections of those lines. The three possibilities are as follows:
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Unique solution: there is only one solution at the intersection of the lines.
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No solution: the lines do not intersect.
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Infinitely many solutions: the lines are identical.
Theorem 4.1.14. Solution of vector equation with invertible matrix.
Consider \(A\vec{x}=\vec{b}\) with \(\mathrm{det}(A)\neq 0\text{.}\) Then there is a unique solution \(\vec{x}=A^{-1}\vec{b}\text{.}\) If also \(\vec{b}=\vec{0}\text{,}\) then \(\vec{x}=A^{-1}\vec{0}=\vec{0}\text{,}\) which is called the trivial solution.
Example 4.1.15.
Let \(\vec{x}\) be a length two vector and let \(c \in \mathbb{R}\text{.}\) Then \(c\vec{x}\) scales the vector’s length, but "preserves" its direction.
Definition 4.1.16. Eigenvalues and eigenvectors.
Let \(A\) be an \(n \times n\) matrix. A constant \(\lambda\) is called an eigenvalue of \(A\) if there is a nonzero vector \(\vec{v}\) such that
\begin{equation*}
A\vec{v}=\lambda \vec{v}.
\end{equation*}
The vector \(\vec{v}\) is called the eigenvector corresponding to the eigenvalue \(\lambda\text{.}\) The pair \(\left\{ \lambda, \vec{v}\right\}\) is called an eigenpair of \(A\text{.}\)
Theorem 4.1.17. Characteristic equation.
Let \(A=\begin{bmatrix}a& b\\c& d\end{bmatrix}\text{.}\) Then the corresponding characteristic equation is \(\lambda^2-\mathrm{tr}(A)\lambda+\mathrm{det}(A)=0.\)
Example 4.1.19.
We rewrite the spring-mass problems as a system of linear ODEs using a substitution. Consider \(my''+\beta y'+ky=f(t)\text{,}\) or in standard form \(y''+2\lambda y+\omega^2 y=u_2(t)\text{.}\)
Example 4.1.20.
Consider two brine tanks, Tank 1 contains \(x(t)\) pounds of salt in \(100\) gallons of brine and Tank 2 contains \(y(t)\) pounds of salt in \(200\) gallons of brine. Each tank is well-stirred and brine in one tank is pumped into the other. In addition, fresh water flows into Tank 1 at a rate of \(20\) gallons per minute and the brine in Tank 2 flows out at a rate of \(20\) gallons per minute so that the total volume of brine remains constant. Find the vector equation of the given system.
Definition 4.1.21. Homogeneous systems.
A linear homogeneous sytem is of the form
\begin{equation}
\vec{x}'=A\vec{x},\tag{4.1.1}
\end{equation}
where \(A\) is an \(n \times n\) matrix.
Definition 4.1.22. Homogeneous solutions.
A vector-valued function \(\vec{w}\colon I \rightarrow \mathbb{R}^n\text{,}\) where \(I\) is the interval of existence, is called a solution to (4.1.1) provided that \(\vec{w}'(t)=A\vec{w}(t)\) for all \(t \in I\text{.}\) Note that the zero vector is always a asolution to (4.1.1), called the trivial solution.
Example 4.1.23.
Let \(\vec{x}'=\begin{bmatrix} 5&-1 \\ 3&1 \end{bmatrix}\vec{x}\text{.}\) Determine if the following are solutions on \((-\infty,\infty)\text{.}\)
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\(\displaystyle \vec{x}=\begin{bmatrix} 1\\ 1 \end{bmatrix} e^{-t}\)
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\(\displaystyle \vec{x}=\begin{bmatrix} 1\\3 \end{bmatrix}e^{2t}\)
Theorem 4.1.24. Nontrivial solutions.
Nontrivial solutions of (4.1.1) are of the form
\begin{equation*}
x_i(t)=\vec{v}_ie^{\lambda_i t},
\end{equation*}
where \(\{\lambda_i,\vec{v}_i\}\) is an eigenpair of \(A\text{.}\)
Theorem 4.1.25. Superposition principle.
Theorem 4.1.26. Linear independence.
The vectors \(\vec{x}_1,\ldots,\vec{x}_n \in \mathbb{R}^n\) are linearly independent on \(I\) if and only if the Wronskian
\begin{equation*}
W(\vec{x}_1,\ldots,\vec{x}_n) = \mathrm{det} \begin{bmatrix} \vec{x}_1 & \vec{x}_2 & \ldots & \vec{x}_n \end{bmatrix} \neq 0.
\end{equation*}
Theorem 4.1.27. Fundamental set of solutions.
The vectors \(\vec{x}_1,\ldots,\vec{x}_n \in \mathbb{R}^n\) form a fundamental set of solutions (FSS) to (4.1.1) on \(I\) if
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\(\vec{x}_1,\ldots,\vec{x}_n\) are linearly independent on \(I\text{,}\) and
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the number of linearly independent solutions is equal to the number of columns of \(A\text{.}\)
Definition 4.1.28. General solution.
Example 4.1.29.
Verify \(\vec{x}_1(t)=\begin{bmatrix} 1\\ 3 \end{bmatrix}e^{2t}\) and \(\vec{x}_2(t)=\begin{bmatrix} 1\\ 1\end{bmatrix}e^{4t}\) form a fundamental set of solutions to \(\vec{x}'=\begin{bmatrix} 5&-1 \\ 3& 1 \end{bmatrix}\vec{x}\) on \((-\infty,\infty)\text{.}\)
Example 4.1.30.
Solve \(\vec{x}'=\begin{bmatrix} 5&-1 \\ 3& 1 \end{bmatrix}\vec{x}, \vec{x}(0)=\begin{bmatrix} 2 \\ -1 \end{bmatrix}.\)
