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The concept of a basis in a vector space is fundamental in understanding dimensions and structure within linear algebra. A basis for a vector space is a set of vectors that fulfills two essential criteria:

• Linear Independence: No vector in the set can be written as a linear combination of the other vectors within the same set.
• Span: The set of vectors must be capable of representing every vector in the vector space through linear combinations.

For example, in our exercise, the set \(S = \{(1,0,0), (1,1,0), (1,1,1)\}\) serves as a basis of \(\mathbb{R}^3\) since it satisfies both criteria by being linearly independent and spanning \(\mathbb{R}^3\). Understanding what a basis is, can help you determine if certain vectors can generate the entire vector space, which provides a clearer insight into multidimensional data representation.

To grasp the concept of span in a vector space, think about the entire set of vectors you can obtain through linear combinations of a given set of vectors. If a set of vectors spans a vector space, then any vector in that space can be represented using the given set.

Consider the vectors in \(S\). They span \(\mathbb{R}^3\) since together, they can produce any vector in \(\mathbb{R}^3\) with the right linear combination. For instance, using the combination of the vectors in \(S\) allows us to express the vector \(\mathbf{u}=(8,3,8)\).

The understanding of the span helps us determine the coverage of our vector space by a set of vectors. It is not enough for vectors to be just linearly independent; they must also span the space to create a complete basis. Consequently, when seeking a basis, you always consider the span as a necessary condition.

The term "linear combination" refers to creating a new vector by multiplying a set of vectors by scalars and then adding the resultant vectors together. If you have vectors \(\mathbf{v}_1, \mathbf{v}_2\), and \(\mathbf{v}_3\) in your set, a linear combination looks like \(a\mathbf{v}_1 + b\mathbf{v}_2 + c\mathbf{v}_3\), where \(a, b,\) and \(c\) are scalars.

In the provided exercise, to express \(\mathbf{u} = (8,3,8)\) as a linear combination of vectors in \(S\), we use the equation \(a(1,0,0) + b(1,1,0) + c(1,1,1) = (8,3,8)\). Solving for \(a, b,\) and \(c\), we find \(a = -3, b = 3,\) and \(c = 8\). This means \(\mathbf{u}\) can specifically be rewritten as \(-3(1,0,0) + 3(1,1,0) + 8(1,1,1)\).

Understanding linear combinations is central to forming different vectors from a given set of vectors, and is critical when dealing with vector representations and transformations.

Reduced Row Echelon Form (RREF) is a key concept in solving systems of linear equations and checking for linear independence among vectors. It involves transforming a matrix to a simplified form using row operations, which makes it easier to understand properties about the vectors or systems represented by the matrix.

In RREF, a matrix looks like this:

• Leading ones: Each leading non-zero entry in a row is 1.
• Zeroes: Each leading 1 is the only non-zero entry in its column.
• Order: All-zero rows, if any, are at the bottom of the matrix.

Returning to our exercise, by arranging the matrix formed by vectors in \(S = \{(1,0,0), (1,1,0), (1,1,1)\}\) into RREF, we confirm their linear independence. The matrix in RREF is:

\[\begin{bmatrix}1 & 1 & 1\0 & 1 & 1\0 & 0 & 1\end{bmatrix}\]
The absence of any zero rows signifies that no vector in \(S\) can be expressed by any combination of the others, proving their independence which is crucial for being a basis. Understanding RREF allows one to solve systems efficiently and verify the relations between vectors.