Checking Basis of Vectors in R3

Check if a set of vectors in $\mathbb{R}^3$ consisting of (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for $\mathbb{R}^3$.

In linear algebra, the concept of a basis is fundamental to understanding vector spaces. A basis of a vector space is a set of vectors that are linearly independent and span the entire space. In this problem, we are tasked with determining if the provided set of vectors forms a basis for three-dimensional real space, often denoted as $\mathbb{R}^3$. The standard basis for $\mathbb{R}^3$ consists of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1), which are the unit vectors along the x, y, and z axes, respectively. These vectors are not only linearly independent, but they also span the entirety of $\mathbb{R}^3$, as any vector in this space can be represented as a unique linear combination of these basis vectors.

When checking if given vectors form a basis in any vector space, we generally consider two main aspects: linear independence and spanning the space. A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. If the set of vectors spans the space, then any vector in the space can be expressed as a combination of the vectors in the set. For the given problem, because the vectors align with the standard basis vectors of $\mathbb{R}^3$, they immediately satisfy both conditions. Therefore, these vectors do form a basis for $\mathbb{R}^3$.

Understanding these properties not only reinforces comprehension of vector spaces but also facilitates the study of linear transformations and the manipulation of coordinate systems, as the ability to change bases is a powerful tool in both theoretical and applied mathematics. Hence, mastering concepts like linear independence and spanning sets builds a strong foundation for more advanced topics such as eigenvectors and diagonalization.