Fundamentals of Linear Algebra

1. Matrix Classifications & Row Echelon Form

A matrix is an ordered rectangular array of numbers arranged into $m$ horizontal rows and $n$ vertical columns, denoted as $A = [a_{ij}]_{m \times n}$.

1.1 Types of Matrices

• Square Matrix: Number of rows equals number of columns ($m = n$).
• Diagonal Matrix: A square matrix where all non-diagonal elements are zero ($a_{ij} = 0$ for all $i \ne j$).
• Upper Triangular Matrix: A square matrix where all elements below the main leading diagonal are zero ($a_{ij} = 0$ for all $i > j$).
• Identity Matrix ($I$): A diagonal matrix whose main diagonal entries are all 1 ($a_{ii} = 1$) and all other entries are zero.
• Symmetric Matrix: A square matrix that equals its transpose: $A = A^T \iff a_{ij} = a_{ji}$

1.2 Row Echelon Form

A non-zero matrix $A$ is in Row Echelon Form if:

1. All completely zero rows are situated below all non-zero rows.
2. The leading non-zero element (pivot) of each row appears strictly to the right of the leading element in the row above it.
3. All entries in the column beneath each leading pivot are zero.

       [ ROW ECHELON FORM ]                    [ RANK DETERMINATION ]
       
       ┌                  ┐                    
       │  [1]   2    3   2│  Row 1 (Non-zero)  
       │   0   [4]   5   1│  Row 2 (Non-zero)  ──► Non-zero rows = 3
       │   0    0   [1]  0│  Row 3 (Non-zero)      Rank ρ(A) = 3
       │   0    0    0   0│  Row 4 (Zero row)  
       └                  ┘                    

1.3 Rank of a Matrix ($\rho(A)$)

The rank of a matrix $A$, denoted by $\rho(A)$, is defined as:

• The order of the largest non-vanishing (non-zero) determinant minor of $A$.
• Equivalent Operational Definition: The total number of non-zero rows remaining in the matrix after reduction to Row Echelon Form via elementary row operations:
  1. $R_i \leftrightarrow R_j$ (Interchanging two rows).
  2. $R_i \to k R_i$ (Multiplying a row by a non-zero scalar $k \ne 0$).
  3. $R_i \to R_i + k R_j$ (Adding a scalar multiple of row $j$ to row $i$).

2. Systems of Linear Equations & Consistency Criteria

A system of $m$ linear equations in $n$ unknowns ($x_1, x_2, \dots, x_n$):