Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Introduction

This tutorial will guide you through the fundamental concepts of linear combinations, span, linear dependence, and basis vectors, as discussed in the video "Linear combinations, span, and basis vectors" from 3Blue1Brown. Understanding these concepts is crucial for anyone studying linear algebra, as they form the foundation for more advanced topics in mathematics and applied sciences.

Step 1: Understand Linear Combinations

• Definition: A linear combination involves taking a set of vectors and multiplying them by scalar coefficients, then summing the results.

• Example: Given vectors v1 and v2, a linear combination could be expressed as:

  c1 * v1 + c2 * v2
  

  where c1 and c2 are scalars.

• Practical Advice:

  • Practice creating linear combinations with simple 2D vectors to visualize how different scalars affect the outcome.
  • Use graphing tools or software to see how varying coefficients changes the resulting vector.

Step 2: Explore the Concept of Span

• Definition: The span of a set of vectors is the collection of all possible linear combinations of those vectors.

• Visual Representation: In a 2D space, the span of two non-parallel vectors forms the entire plane, whereas the span of a single vector forms a line.

• Practical Advice:

  • Check if a vector is in the span of a set by trying to express it as a linear combination of the vectors in that set.
  • Use geometric interpretations to better understand how vectors can combine to fill spaces.

Step 3: Learn About Linear Dependence and Independence

• Linear Dependence: A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of the others.

• Linear Independence: A set is independent if no vector can be expressed in terms of the others.

• Practical Advice:

  • To test for dependence, try to find scalars (not all zero) such that:

  c1 * v1 + c2 * v2 + ... + cn * vn = 0
  

  If you can find such scalars, the vectors are dependent.

  • Use the concept of rows in a matrix to visualize dependence; if one row can be rewritten using the others, they are dependent.

Step 4: Understand Basis Vectors

• Definition: A basis is a set of vectors that are linearly independent and span a vector space.

• Key Properties:

  • The number of vectors in a basis corresponds to the dimension of the space.
  • Every vector in the space can be uniquely represented as a linear combination of the basis vectors.

• Practical Advice:

  • Identify a basis for simple vector spaces like R² or R³ by selecting linearly independent vectors.
  • Experiment with different sets of vectors to see which can serve as a basis for a given space.

Conclusion

In this tutorial, you've explored linear combinations, the concept of span, linear dependence and independence, and basis vectors. These foundational ideas are essential for deeper exploration in linear algebra. To further solidify your understanding, practice by working with different sets of vectors and visually representing their spans and dependencies through graphs. Consider applying these concepts in real-world scenarios like computer graphics, data analysis, or physics simulations.