Subspaces Basis and Dimension

<p data-id="1">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.</p><p data-id="2">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 $\mathbb{R}^3$.</p><p data-id="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.</p><div data-youtube-video=""><iframe allowfullscreen endTime="0" ivLoadPolicy="0" origin="" playlist="" src="https://www.youtube.com/embed/4C9GKyfUQkc?start=151" start="151"></iframe></div>

Checking Basis of Vectors in R3

<p data-id="1">When determining if a subset of a vector space is a subspace, it is crucial to check for two fundamental properties: closure under addition and closure under scalar multiplication. These properties ensure that any linear combination of the vectors within the subset remains within the subset, maintaining the structural integrity of a vector space. For the subset given in this problem, which includes vectors of the form (x, 0, -x), consider how any two such vectors are added and how they are affected by scalar multiplication. This involves proving that for any vectors u and v in the subset, as well as any scalar c, the vector cu is also in the subset and so is u + v. This analysis reinforces the importance of understanding subspace properties and highlights the elegance of linear algebra in defining and manipulating vector spaces. Essentially, you are establishing an internal consistency that aligns with the vector space axioms, ensuring that the subset behaves like a vector space within the larger vector space V.</p><div data-youtube-video=""><iframe allowfullscreen endTime="0" ivLoadPolicy="0" origin="" playlist="" src="https://www.youtube.com/embed/tM4TDL9Hj8U?start=140" start="140"></iframe></div>

Linear Algebra: Subspaces Basis and Dimension