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The row echelon form (REF) of a matrix is a simplified version that helps solve systems of linear equations, making it easier to perform calculations and understand the matrix's properties. To transform a matrix into this form, we perform elementary row operations. In REF:

• All non-zero rows are above any rows of all zeroes.
• The leading entry of each non-zero row after the first appears to the right of the leading entry of the preceding row.
• The leading entry in any row is 1, sometimes called a leading 1.

Transforming a matrix into row echelon form involves making strategic use of row operations: swapping rows, multiplying rows by constants, and adding multiples of one row to another. These operations do not change the solutions of the system of equations represented by the matrix. Once a matrix is in REF, it becomes considerably easier to perform back-substitution when solving equations.

Nonsquare matrices are matrices that do not have an equal number of rows and columns, formally called non-square. In mathematical terms, a nonsquare matrix is of the form \( m \times n \), where the number of rows \( m \) differs from the number of columns \( n \).

When performing LU factorization on a nonsquare matrix, we modify our approach slightly. Instead of requiring the matrix \( U \) in the factorization to be an upper triangular matrix (as in square matrices), we require it to be in row echelon form, which provides a structurally stable way to deal with the additional or fewer dimensions that come with nonsquare matrices. This helps cater to the various constraints imposed by linear transformations that aren't captured by square matrices alone.

Understanding nonsquare matrices is crucial in various applications like solving systems arising in real-world problems where inputs or outcomes do not necessarily align in perfect squares, such as in statistics or network analysis, where one might have more variables than equations or vice versa.

A lower triangular matrix is a type of square matrix where all entries above the main diagonal are zero. In symbol form, for an \( n \times n \) matrix \( L \), if \( i < j \), then \( L_{ij} = 0 \), where \( i \) and \( j \) are indices of row and column, respectively.

This type of matrix plays a crucial role in LU factorization, as it captures the row operations through multipliers applied during the transformation of a matrix to its row echelon form. Essentially, each non-zero entry below the diagonal corresponds to a multiplier that was used to achieve zeros in the elimination process.

In practical applications, identifying or calculating with lower triangular matrices can simplify computations significantly. For example, matrix inversion or solving equations is computationally faster when dealing with triangular matrices due to their reduced number of non-zero elements.