Week 7: Dimensions and Linear Transformations

Week 7 Summary

This week we explored three fundamental concepts in linear algebra that build upon our understanding of vector spaces:

1. Dimensions of Vector Spaces

• The concept of dimension as the size of a basis
• The key lemma relating linearly independent sets and spanning sets

2. Linear Transformations

• Definition and properties of linear transformations
• Matrix representations of linear transformations
• Composition of linear transformations

3. Null Spaces and Kernels

• Definition of null space (kernel) of a linear transformation
• Relationship between null space and solutions to homogeneous systems

Key Lemma

One of the most important results we covered is:

Key Lemma: Let V be a vector space, and suppose that {u₁, u₂, ..., uₐ} is a set of linearly independent vectors in V, and that {v₁, v₂, ..., vᵦ} is a spanning set for V. Then a ≤ b.

This lemma establishes that:

• Any linearly independent set can have at most as many vectors as any spanning set
• All bases of a vector space have the same number of elements
• The dimension of a vector space is well-defined