Linear Equations in Linear Algebra

Intro to Linear Equations in Linear Algebra Notation Dot Product vs Inner Product and Column Vectors

This lesson explains to students in detail the difference between vector notations in linear algebra and what to expect with terminology.

This is a system of linear equations. A linear equation is of the form

$ax = b, \quad ax + by = c, \quad ax + by + cz = d$

In differential equations, a linear equation equates to derivatives not being operated on.

E.g., A linear-nonhomogeneous ODE

$ay''' + by'' + cy' + dy = x^2$

$ax + by = c, \quad dx + ey = f$

Let’s think about the dot product

$ax + by = \langle a, b \rangle \cdot \langle x, y \rangle = c, \quad dx + ey = \langle a, b \rangle \cdot \langle x, y \rangle = f$

Note* Beyond Calc 3, we don’t use $\langle \dots \rangle$ for points/vectors; we use $\begin{pmatrix} \dots \\ \dots \end{pmatrix}$ (columns). All vectors are column vectors unless transposed or otherwise stated.

Let’s get ahead of ourselves and look at the inner product and column vector notations

Note* $(a, b)$ is not a row vector. A row vector is $(a, b)^T = \begin{pmatrix} a \\ b \end{pmatrix}^T \equiv [a \ b]$

$ax + by = (a, b) \cdot \langle x, y \rangle = c, \quad dx + ey = (d, e) \cdot \langle x, y \rangle = f$

$ax + by = \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = c, \quad dx + ey = \begin{pmatrix} c \\ d \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = f \quad \text{[dot product]},$

$ax + by = \begin{pmatrix} a \\ b \end{pmatrix}^T \begin{pmatrix} x \\ y \end{pmatrix} = c, \quad dx + ey = \begin{pmatrix} c \\ d \end{pmatrix}^T \begin{pmatrix} x \\ y \end{pmatrix} = f \quad \text{[inner product]},$

$ax + by = (a, b) \begin{pmatrix} x \\ y \end{pmatrix} = c, \quad dx + ey = (c, d) \begin{pmatrix} x \\ y \end{pmatrix} = f \quad \text{[inner product]}.$

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c \\ f \end{pmatrix} \Rightarrow \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} f \end{bmatrix} \Rightarrow Ax = b$

Linear equations in linear algebra represent systems of equations where each equation is linear, involving variables with no powers higher than one, and constants. These systems can be expressed in the form:

$a_1x_1 + a_2x_2 + \dots + a_nx_n = b$

where:

$a_1, a_2, \dots, a_n$ are coefficients,
$x_1, x_2, \dots, x_n$ are variables, and
$b$ is the constant term.

In matrix form, a system of $m$ equations with $n$ variables can be written as:

$A \mathbf{x} = \mathbf{b}$

where:

$A$ is the $m \times n$ coefficient matrix,
$\mathbf{x}$ is the column vector of variables,
$\mathbf{b}$ is the column vector of constants.

Example

Given a system of equations:

$2x + 3y = 5$

$4x - y = 6$

It can be expressed in matrix form:

$\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}.$

Key Concepts:

Solution Sets:
- Unique Solution: If the system has exactly one solution.
- Infinite Solutions: If the equations are dependent and consistent.
- No Solution: If the system is inconsistent.
Row Reduction: Systems can be solved using Gaussian elimination or row reduction to reduced row echelon form (RREF).
Applications: Linear equations in linear algebra are foundational for solving real-world problems in engineering, physics, computer science, and economics.

Let me know if you’d like detailed examples, practice problems, or explanations of related topics like matrix operations or determinants!

Linear Equations in Linear Algebra

Intro to Linear Equations in Linear Algebra Notation Dot Product vs Inner Product and Column Vectors