Introduction: Building Blocks of a Space
Not all vectors in a vector space are equally important. Some vectors can be written as combinations of others; they're redundant. The concepts of linear independence and span help us identify the essential building blocks of any vector space.
Linear Combinations
A linear combination of vectors \(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\) is any expression of the form:
\[c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n\]where the \(c_i\) are scalars from the field.
Span: What Can You Reach?
The span of a set of vectors is the collection of all possible linear combinations of those vectors:
\[\text{span}\{\vec{v}_1, \ldots, \vec{v}_n\} = \{c_1\vec{v}_1 + \cdots + c_n\vec{v}_n \mid c_i \in \mathbb{F}\}\]If the span equals the entire vector space, we say the vectors span the space.
Linear Independence: No Redundancy
A set of vectors \(\{\vec{v}_1, \ldots, \vec{v}_n\}\) is linearly independent if the only solution to:
\[c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n = \vec{0}\]is \(c_1 = c_2 = \cdots = c_n = 0\). If any other solution exists, the vectors are linearly dependent.
Worked Examples
Example 1: Testing Independence in \(\mathbb{R}^2\)
Are \(\vec{v}_1 = (1, 2)\) and \(\vec{v}_2 = (3, 6)\) linearly independent?
Set up: \(c_1(1, 2) + c_2(3, 6) = (0, 0)\)
This gives: \(c_1 + 3c_2 = 0\) and \(2c_1 + 6c_2 = 0\)
Notice \(\vec{v}_2 = 3\vec{v}_1\), so \(c_1 = 3, c_2 = -1\) is a non-trivial solution.
Result: These vectors are linearly dependent.
Example 2: Independent Vectors in \(\mathbb{R}^2\)
Are \(\vec{v}_1 = (1, 0)\) and \(\vec{v}_2 = (0, 1)\) linearly independent?
\(c_1(1, 0) + c_2(0, 1) = (0, 0)\) gives \(c_1 = 0\) and \(c_2 = 0\).
Result: Only the trivial solution exists—they are linearly independent.
Example 3: Span in \(\mathbb{R}^3\)
What is span\(\{(1, 0, 0), (0, 1, 0)\}\)?
Any linear combination gives \((a, b, 0)\)—the \(xy\)-plane in 3D space. These two vectors span a 2D subspace of \(\mathbb{R}^3\).
The Quantum Connection
In quantum mechanics, we often expand a state \(|\psi\rangle\) as a linear combination of basis states. The requirement that basis states be linearly independent ensures each quantum state has a unique expansion. If states were dependent, we'd have ambiguous descriptions of reality—a mathematical disaster for physics.