University Linear Algebra Worksheets: Vector Spaces and Subspaces

📚 Topic Summary

In linear algebra, a vector space is a set of objects (vectors) that can be added together and multiplied by scalars. A subspace is a subset of a vector space that is itself a vector space under the same operations. To prove a subset is a subspace, you must show it contains the zero vector, is closed under addition, and is closed under scalar multiplication.

Understanding vector spaces and subspaces is crucial for solving linear equations, understanding transformations, and many other applications in mathematics, physics, and computer science.

🧮 Part A: Vocabulary

Match the following terms with their definitions:

1. Term: Vector Space
2. Term: Subspace
3. Term: Linear Combination
4. Term: Span
5. Term: Basis

Definitions:

1. A set of vectors that can be added and scaled.
2. A subset of a vector space that is also a vector space.
3. A set of vectors that generates the entire vector space.
4. A sum of scalar multiples of vectors.
5. A linearly independent set that spans the vector space.

✍️ Part B: Fill in the Blanks

A _________ is a subset of a vector space that is closed under addition and scalar multiplication, and contains the _________ vector. The _________ of a set of vectors is the set of all their linear combinations. A _________ is a set of linearly independent vectors that spans the entire vector space, and its size is the _________ of the vector space.

🤔 Part C: Critical Thinking

Explain why the set of all solutions to a homogeneous system of linear equations, $Ax = 0$, forms a subspace of $\mathbb{R}^n$.