Linear Algebra: Matrices and Vectors

Study Notes

Chapter 1: Introduction

A vector is an ordered list of numbers:

$$x = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix}$$

$x_1, x_2, ..., x_n$ are the vector's components.
A matrix is a rectangular array of numbers:

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

$a_{ij}$ is the element in the i-th row and j-th column.
Vector Addition: $x + y = [x_1 + y_1, x_2 + y_2,..., x_n + y_n]^T$
Scalar Multiplication: $\alpha x = [\alpha x_1, \alpha x_2,..., \alpha x_n]^T$
Matrix Product: $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$
The transpose of matrix A, denoted as $A^T$, is obtained by interchanging rows and columns.
Square Matrix: Number of rows equals the number of columns.
Identity Matrix: A square matrix with 1s on the diagonal and 0s elsewhere.
Diagonal Matrix: A square matrix with non-zero elements only on the diagonal.

Chapter 2: Systems of Linear Equations

A system of linear equations is a set of equations in the form:

$a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1$

$a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2$...

$a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m$

Methods to solve systems of linear equations include:
Gaussian elimination
Substitution
The inverse matrix method
Linear equations have applications in electrical networks, economics, and engineering.