91Ó°ÊÓ

In the context of vector spaces, a spanning set is a collection of vectors that can be combined in various ways to form any vector in the space they span. If a set of vectors spans a vector space, it can generate any vector in that space through linear combinations. For example, in three-dimensional space, we often work with the standard basis vectors that span the entirety of the space.
In the given exercise, the set \(S = \{(1,0,1), (1,1,0), (0,1,1)\}\) consists of vectors in \(\mathbb{R}^3\). To determine if these vectors span \(\mathbb{R}^3\), we check if any arbitrary vector \((x, y, z)\) in \(\mathbb{R}^3\) can be expressed as a linear combination of these vectors. Unfortunately, not all such vectors \((x, y, z)\) can be created from \(S\), so it does not span \(\mathbb{R}^3\). Instead, it spans a subspace within it, specifically a plane in this three-dimensional space.

Linear combinations involve combining vectors using scalar multiplication and addition. In more formal terms, if you have vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\) and scalars \(a_1, a_2, \ldots, a_n\), a linear combination is any vector that can be formed as \[a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n\]Linear combinations are crucial for determining spanning sets because they show how elements of the set can be mixed to form new vectors.
In the exercise, we expressed a vector \((x, y, z)\) as a linear combination of the vectors in set \(S\). This helped check if \(S\) spans \(\mathbb{R}^3\). The resulting system of equations gives a clear picture of how the coefficients \(a, b, c\) relate to any desired vector in the space:

• \(a + b = x\)
• \(b + c = y\)
• \(a + c = z\)

Upon solving these equations, some vectors won't align perfectly, illustrating the limitations of the set and showing that they don't form \(\mathbb{R}^3\). It highlights the criticality of checking each component's contribution to forming any vector in the space.

Subspaces are an essential concept in linear algebra and vector spaces. They refer to a set of vectors within a vector space that itself complies with the rules of vector spaces. That means a subspace must be closed under vector addition and scalar multiplication. So, essentially, a subspace is a smaller vector space contained within a larger one.
In our exercise, though the set \(S = \{(1,0,1), (1,1,0), (0,1,1)\}\) does not span the full space of \(\mathbb{R}^3\), it forms a subspace. Since these vectors are not all coplanar, the set \(S\) spans a plane through the origin in \(\mathbb{R}^3\). This plane is made of all possible linear combinations of the vectors in \(S\). This subspace is significant because it allows us to understand which part of \(\mathbb{R}^3\) is covered by these vectors. It is essentially a two-dimensional surface within the larger three-dimensional space.