📚 Understanding Span vs. Linear Independence
Span and linear independence are fundamental concepts in linear algebra. While they're related, they describe different aspects of vector spaces. Span deals with the set of all possible linear combinations, while linear independence focuses on whether a set of vectors can be combined to produce the zero vector in a non-trivial way.
📌 Definition of Span
The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it's everything you can reach by adding and scaling the vectors in your set.
📐 Definition of Linear Independence
A set of vectors is linearly independent if the only way to get the zero vector by taking a linear combination of them is to multiply all the vectors by zero. If there's another way to get the zero vector, the vectors are linearly dependent.
📝 Span vs. Linear Independence: A Detailed Comparison
| Feature |
Span |
Linear Independence |
| Definition |
The set of all possible linear combinations of a set of vectors. |
A set of vectors is linearly independent if the only linear combination that equals the zero vector is the trivial one (all coefficients are zero). |
| Focus |
What space can be reached using the vectors. |
Whether the vectors are truly distinct and contributing unique information. |
| Zero Vector |
The span always contains the zero vector (since multiplying all vectors by zero gives the zero vector). |
Linear independence is determined by whether there is a non-trivial linear combination that results in the zero vector. |
| Redundancy |
A spanning set can contain redundant vectors (vectors that can be expressed as a linear combination of the others). |
Linearly independent sets contain no redundant vectors. |
| Example |
The span of $\{(1, 0), (0, 1)\}$ is $\mathbb{R}^2$ (the entire 2D plane). |
The set $\{(1, 0), (0, 1)\}$ is linearly independent. |
| Implication |
If the span of a set of vectors is the entire vector space, the set is a spanning set for that vector space. |
If a set of vectors is linearly independent, then no vector in the set can be written as a linear combination of the other vectors in the set. |
🔑 Key Takeaways
- 📍 Span: Describes the space that a set of vectors can "reach".
- 🧭 Linear Independence: Checks if vectors provide unique directions and are not redundant.
- ➕ Connection: A basis of a vector space is a set of linearly independent vectors that span the entire space. It's the perfect combination of efficiency (linear independence) and coverage (span).