Basis is a crucial concept in linear algebra, as it provides a way to understand vector spaces and express vectors within them. Essentially, a basis is a set of linearly independent vectors that span a vector space. Let’s break down this definition to better understand the concept of a basis:
1. *Linearly Independent Vectors*: A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the other vectors in the set. In other words, no single vector is "redundant" and can’t be formed by adding or subtracting other vectors in the set with appropriate scalar weights.
2. *Span a Vector Space*: A set of vectors spans a vector space if every vector in the vector space can be expressed as a linear combination of the vectors in the set.
A basis is a combination of these two properties. Any set of vectors that are linearly independent and span the vector space form a basis for that space. It is important to note that a basis is not unique, and there can be multiple bases for the same vector space.
Let’s consider an example in the two-dimensional Euclidean space, also known as ℝ2. Here, the standard basis is given by the set {e1, e2}, where $\mathbf{e_1} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\mathbf{e_2} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$. The standard basis is linearly independent, as neither vector can be formed by scaling or adding the other vector. Furthermore, every vector in ℝ2 can be expressed as a linear combination of these two vectors.
For example, take the vector $\mathbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$. Using the standard basis vectors, we can express the vector as follows:
$$\mathbf{v} = 3\mathbf{e_1} + 4\mathbf{e_2} = 3\begin{bmatrix} 1 \\ 0 \end{bmatrix} + 4\begin{bmatrix} 0 \\ 1 \end{bmatrix}$$
As mentioned, a basis is not unique. Another basis for the vector space ℝ2 could be the set {a, b}, where $\mathbf{a} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$. These vectors are linearly independent, and every vector in ℝ2 can be formed as a linear combination of a and b.