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Understanding Gaussian elimination is crucial for solving systems of linear equations, as well as determining the span of a set of vectors. Essentially, it's a method you can use to simplify a matrix to discover important characteristics about the vector set it represents.

When you start with the matrix representation of a set of vectors, Gaussian elimination helps you perform a series of operations, such as swapping rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another. The goal? To reach what's known as row echelon form or even reduced row echelon form, where the structure of the matrix reveals whether the vector set can span the entire space or a subspace.

By carefully applying these steps – the key operations of Gaussian elimination – you can unveil the dependencies between the vectors. If a vector is found to be a combination of others, it doesn't contribute to expanding the dimensions of the space that can be spanned; therefore, understanding whether a set of vectors has the potential to span an entire space like \( R^3 \) is rooted in this elimination process.

Upon applying Gaussian elimination, you aim to get your matrix into row echelon form (REF). But what is REF and why is it so important in linear algebra? A matrix is in row echelon form when all non-zero rows are above any rows of all zeros, and each leading coefficient of a row is to the right of the leading coefficient of the row above it.

Characteristics of REF

• Non-zero rows are at the top.
• Leading entries (first non-zero number from the left, in a row) go down and to the right.
• Rows with all zeros are at the bottom.

Reaching REF is like having a 'skeleton' of your matrix: it’s easier to see the basic structure without all the extra 'flesh' of complicated numbers. Essentially, it provides a simpler form to understand which vectors in your set are independent and, thereby, determine the actual dimensionality of the space spanned by them.

The rank of a matrix is a fundamental concept that tells you the maximum number of linearly independent row or column vectors in the matrix. Why is this important? Because the rank gives you the dimension of the vector space spanned by these vectors.

When you perform Gaussian elimination and get your matrix into row echelon form, the number of non-zero rows left gives you the matrix rank. So, if your aim is to determine whether a set of vectors spans \( R^3 \), you would want your matrix to have a rank of 3. This means that all three vectors are linearly independent and collectively have the ability to reach every corner of \( R^3 \), making no corner inaccessible. A rank less than 3 would reveal that the vectors only span a subspace of \( R^3 \), such as a plane or a line, depending on whether the rank is 2 or 1, respectively.

So what happens if a set of vectors doesn’t span the entirety of \( R^3 \)? They define what's called a subspace. A subspace is simply a smaller space within the larger space that the vectors can completely cover. In \( R^3 \), the possible subspaces are planes or lines — or just the trivial case, a single point at the origin if all vectors are zero.

If your matrix has a rank of 2 after Gaussian elimination, it means the vectors span a plane within \( R^3 \). In this plane, you can reach any point using combinations of just those vectors. If the rank is 1, they span a line. Understanding the geometric subspace that a set of vectors spans is crucial for visualizing solutions to systems of equations and for grasping how vectors interact in multi-dimensional spaces.