Key Concepts of Vector Spaces to Know for Linear Algebra 101

Vector spaces are fundamental in linear algebra, representing collections of vectors that can be added and scaled. Understanding their structure, including subspaces, linear independence, and transformations, is key for solving equations and analyzing mathematical systems in both linear algebra and differential equations.

1. Definition of a vector space

   • A vector space is a collection of vectors that can be added together and multiplied by scalars.
   • It must satisfy eight axioms, including closure under addition and scalar multiplication.
   • Examples include (\mathbb{R}^n) and function spaces.

2. Subspaces

   • A subspace is a subset of a vector space that is also a vector space itself.
   • It must contain the zero vector, be closed under addition, and be closed under scalar multiplication.
   • Common examples include lines and planes through the origin in (\mathbb{R}^3).

3. Linear independence and dependence

   • A set of vectors is linearly independent if no vector can be expressed as a linear combination of the others.
   • If at least one vector can be expressed as a combination of others, the set is linearly dependent.
   • Linear independence is crucial for determining the basis of a vector space.

4. Basis and dimension

   • A basis is a set of linearly independent vectors that spans the vector space.
   • The dimension of a vector space is the number of vectors in a basis for that space.
   • Dimension provides insight into the "size" of the vector space.

5. Span of vectors

   • The span of a set of vectors is the set of all possible linear combinations of those vectors.
   • It represents all the vectors that can be reached using the given vectors.
   • The span can be finite or infinite, depending on the number of vectors.

6. Linear transformations

   • A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication.
   • It can be represented by a matrix, which simplifies calculations.
   • Linear transformations are essential for understanding how vector spaces interact.

7. Null space and range

   • The null space of a linear transformation consists of all vectors that map to the zero vector.
   • The range is the set of all possible outputs of the transformation.
   • Understanding both helps in analyzing the properties of linear transformations.

8. Eigenvalues and eigenvectors

   • An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied.
   • The corresponding scalar is called the eigenvalue.
   • Eigenvalues and eigenvectors are critical for solving systems of differential equations and understanding matrix behavior.

9. Inner product spaces

   • An inner product space is a vector space equipped with an inner product, which defines a notion of angle and length.
   • The inner product allows for the generalization of concepts like orthogonality and distance.
   • Common examples include Euclidean spaces with the standard dot product.

10. Orthogonality and orthonormal bases

    • Vectors are orthogonal if their inner product is zero, indicating they are at right angles to each other.
    • An orthonormal basis consists of orthogonal vectors that are also unit vectors (length of 1).
    • Orthonormal bases simplify calculations and are useful in various applications, including projections and decompositions.