📚 Definition of Span in $R^n$
In linear algebra, the span of a set of vectors in $R^n$ (where $R$ represents the real numbers and $n$ is a positive integer representing the dimension) is the set of all possible linear combinations of those vectors. Essentially, it's the set of all vectors that can be created by scaling and adding together the original vectors.
📜 History and Background
The concept of span grew out of the need to understand vector spaces and subspaces more formally. Early work on linear equations and matrices in the 18th and 19th centuries gradually led to the development of abstract vector spaces and the idea that sets of vectors could 'generate' these spaces. Mathematicians like Arthur Cayley and Hermann Grassmann played significant roles in formulating these ideas.
📌 Key Principles
- ➕ Linear Combination: A linear combination of vectors $v_1, v_2, ..., v_k$ in $R^n$ is an expression of the form $c_1v_1 + c_2v_2 + ... + c_kv_k$, where $c_1, c_2, ..., c_k$ are scalars (real numbers).
- 🎯 Span Definition: The span of a set of vectors $S = \{v_1, v_2, ..., v_k\}$ in $R^n$, denoted as $span(S)$, is the set of all possible linear combinations of the vectors in $S$. Mathematically, $span(S) = \{c_1v_1 + c_2v_2 + ... + c_kv_k : c_1, c_2, ..., c_k \in R\}$.
- 📐 Geometric Interpretation: In $R^2$, the span of a single non-zero vector is a line through the origin. The span of two non-collinear vectors in $R^2$ is the entire plane $R^2$. In $R^3$, the span of a single non-zero vector is a line, the span of two non-collinear vectors is a plane, and the span of three non-coplanar vectors is the entire space $R^3$.
- ✨ Subspace: The span of any set of vectors in $R^n$ is always a subspace of $R^n$. This means it contains the zero vector, and is closed under addition and scalar multiplication.
🌍 Real-World Examples
Example 1: Span in $R^2$
Let $v_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$.
Then $span(\{v_1, v_2\})$ is the entire $R^2$ plane, because any vector $\begin{bmatrix} x \\ y \end{bmatrix}$ in $R^2$ can be written as $x\begin{bmatrix} 1 \\ 0 \end{bmatrix} + y\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.
Example 2: Span in $R^3$
Let $v_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$.
Then $span(\{v_1, v_2\})$ is the $xy$-plane in $R^3$. Any vector in the $xy$-plane can be written as a linear combination of $v_1$ and $v_2$.
📝 Conclusion
Understanding the span of a set of vectors is crucial in linear algebra. It helps to define vector spaces, subspaces, and the concepts of linear independence and basis. It also allows us to describe the range of linear transformations, making it a vital tool for various applications in mathematics, physics, engineering, and computer science.