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Linear independence is a fundamental concept in vector spaces that determines whether a set of vectors can be expressed as a linear combination of the others. To determine if the vectors in set S are linearly independent, no vector in the set should be expressible as a combination of the others. In mathematical terms, for vectors v1, v2, ..., vn to be linearly independent, the equation c1*v1 + c2*v2 + ... + cn*vn = 0 must have only one solution, which is c1 = c2 = ... = cn = 0.

For our matrix A, if we can't find non-zero coefficients c1, c2, c3, and c4 that satisfy the equation c1*(1,0,0,1) + c2*(0,2,0,2) + c3*(1,0,1,0) + c4*(0,2,2,0) = (0,0,0,0) other than c1 = c2 = c3 = c4 = 0, then the vectors are linearly independent. This characteristic is crucial for a set to be considered a basis for a vector space, as each basis vector must contribute uniquely to the vector space structure.

Gaussian elimination is a method used to solve systems of linear equations and also to understand the properties of a matrix. It involves a series of operations to transform a given matrix into a row-echelon form. These operations include swapping rows, multiplying a row by a non-zero constant, and adding or subtracting rows from each other.

While working on set S, these operations start by creating zeros below the pivots - the leading non-zero element in a row. The goal is to wind up with a triangular form where the lower left corner of the matrix consists entirely of zeros. Gaussian elimination provides an effective way to determine the linear independence of a set of vectors by revealing a clear upper-triangle structure in the matrix, which is the row-echelon form we are targeting.

Row-echelon form is a specific arrangement of a matrix where each row has a leading non-zero element, and these leading entries shift to the right as you move down the rows. Furthermore, all the entries below a pivot must be zeros. Achieving row-echelon form is an essential step in solving linear equations and determining the properties of a matrix.

Applying Gaussian elimination to our matrix A, the result is a diagonal matrix which is a specialized type of row-echelon form where all non-zero entries are along the diagonal. This structure immediately tells us that the vectors are linearly independent and provides a valuable insight into the vector span, as each row now corresponds to a unique direction in the vector space.

Vector span is the set of all possible vectors that can be created through linear combinations of a set of vectors. In simpler terms, if you have a set of vectors, any vector in the vector space can be represented as a combination of those in the set. A basis for a vector space is a set of vectors that span the space while also being linearly independent.

In the context of our set S, checking if the vectors span R^4 is equivalent to confirming that any 4-dimensional vector can be formed using a linear combination of the vectors in S. Since we have four vectors, and after applying Gaussian elimination the matrix ranks as 4, it confirms that our set S spans R^4. Hence, S is indeed a basis for the vector space R^4, as it satisfies both conditions of spanning R^4 and being linearly independent.