1.1
Matrices and Systems of Equations
A differential equation involves the derivative of an unknown function. THe following examples are very simple applications of differential equations.
- \( y'=2x \) \[\begin{align} y&=\int 2x \\ &=x^2 + c \end{align}\]
- \( y'=ky \) (exponential growth and decay equation) \[\begin{align} y&=Ce^{kx} \end{align}\]
Linear algebra is the study of systems of linear equations. In two variables, this means that we're looking at lines and where they intersect. In three variables, this means that we're considering planes and where they intersect.
We can solve two equations using the traditional substitution method, but this can become unwieldy and challenging with more complex problems. Instead, we'll opt to use and consider the elimination method.
- \( \begin{cases} 2x+3y=6 \\ 5x-y=4 \end{cases} \) \[\begin{align} (2x+3y=6)&+3(5x-y=4) \\ 17x+0y&=18 \\ x&=\frac{18}{17} \end{align}\]
We can keep track of coefficients using matrices. Let's continue the previous example.
- \( \begin{cases} 2x+3y=6 \\ 5x-y=4 \end{cases} \) \[ \left[ \begin{matrix} x & y \\ 2 & 3 \\ 5 & -1 \\ \end{matrix} \left| \, \begin{matrix} c \\ 6 \\ 4 \\ \end{matrix} \right. \right] \]
A matrix is an \( (m\times n) \) array of elements formed by \(n\) rows and \(m\) columns. It is typically denoted by a capital letter, like \(A\), while its elements are referred to by lowercase letters with their index, like \(a_{ij}\), where \(i\) and \(j\) refer to the row and column.
Matrices are only equal if they are exactly the same. In other words, \(A=B\) if and only if \(A\) and \(B\) are \( (m\times n) \) and \( a_{ij}=b_{ij}\) for all \(i\) and \(j\).
A row vector is a \((1\times n)\) matrix, while a column vector is a \((m\times 1)\) matrix.
The transpose of a matrix effectively swaps the rows and columns. It is denoted by a superscript T to the matrix letter.
- What is the transpose of \( A=\begin{bmatrix} 1 & 3 & 5 & 7 \\ 2 & 4 & 6 & 8 \\ \end{bmatrix} \) \[ A^T=\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ 7 & 8 \\ \end{bmatrix} \]
Notice that when the indexes are repeated, like \(a_{ii}\), the element will not actually change positions.
The elements \(a_{ii}\) are located along the main diagonal, which starts at the top left corner of a matrix.
A matrix is square if \(n=m\) in a \( (n\times n) \) matrix.
A matrix is lower triangular if all elements above the main diagonal are zero. A matrix is upper triangular if all elements below the main diagonal are zero.
A matrix is diagonal if it is upper and lower triangular, and \(a_{ij}=0\) for \(i\neq j\).
A \((n\times n)\) square matrix can be symmetric about the diagonal, meaning that it is equal to its transpose. A \((n\times n)\) square matrix can be anti-or-skew-symmetric if it is equal to its negative transpose and if the diagonal is zero.
A matrix function has functions as its elements instead of numbers (or constant functions).