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Gaussian elimination is a systematic method used for solving linear systems and finding the rank of a matrix. It is also instrumental in determining the basis of a subspace. The method involves a series of operations applied to the original matrix: swapping rows, multiplying a row by a nonzero number, and adding or subtracting a multiple of one row from another row.

The purpose of these operations is to transform the matrix into an upper triangular form, where all elements below the diagonal are zeros. This form is known as the row echelon form. If you continue the process to make all elements above the leading coefficients zero as well, you'll reach the reduced row echelon form. This is the form that reveals the basis of the subspace spanned by the rows of the original matrix, as each pivot (leading 1) corresponds to a dimension of the subspace.

The row echelon form of a matrix is an intermediate step in the Gaussian elimination process. It is identified by several key features: each nonzero row has a leading 1 that is to the right of the leading 1 in the row above, known as a pivot. All entries in a column below a pivot are zeros. If there are any rows containing only zeros, they are at the bottom of the matrix.

To get a matrix into this form, you start by getting a leading 1 in the top row, then use row operations to get zeros below it. You repeat this step for each pivot. The row echelon form is critical as it simplifies solving linear systems and is the first step in finding the basis for a subspace because it helps identify linearly independent vectors that span the space.

In linear algebra, the span of a set of vectors is the set of all possible linear combinations of those vectors. Mathematically, if you have a set of vectors \( v_1, v_2, ..., v_k \), the span is the set \( \{ a_1v_1 + a_2v_2 + ... + a_kv_k | a_i \in \mathbb{R} \} \).

This concept is integral to understanding vector spaces and subspaces. A basis for a subspace is a set of vectors in the subset that are linearly independent and span the subspace. When finding a basis for a subspace, you're essentially looking for the smallest set of vectors that can be combined in various ways to reach any point within that subspace.

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. Linear algebra is fundamental to many areas of mathematics and its applications, including engineering, natural science, social sciences, and applied mathematics.

Key concepts in linear algebra include vectors, matrices, determinants, span, basis, and dimension. It's essential for solving systems of linear equations, performing transformations, and working with structures that capture the linear properties of operations such as rotation and scale.