MAS295 Vector Spaces (NJTech)

Semester 2, 2019/20 10 Credits
Lecturer: Dr Fionntan Roukema Reading List
Aims Outcomes Assessment Full Syllabus

This course introduces abstract vector spaces and linear transformations, building on the concrete approach with matrices in earlier courses. Many results that were merely stated, or proved by matrix methods, will be given conceptual proofs in this course. The abstract approach will allow us to give efficient proofs that simultaneously tell us interesting things about vectors, matrices, polynomials, sequences, differential equations, and many other objects. A central aim of the course is to help students become comfortable with the required level of abstraction.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

Outline syllabus

  • Vector spaces, linear maps, subspaces.
  • Independence and spanning sets.
  • Linear maps out of Rn; matrices for linear maps.
  • Theorems about bases.
  • Eigenvalues and eigenvectors.


  • Introduce the abstract theory of vector spaces and linear maps between them.
  • Introduce the abstract theory of inner product spaces.
  • Familiarise students with abstract and axiomatic mathematics.

Learning outcomes

  • Students should understand the main concepts of linear algebra: vector spaces; subspaces, (direct) sum and intersection; linear maps, kernels and images; independent sets, spanning sets and bases; relations with matrices.
  • Students should be familiar with a range of examples of these concepts.
  • Students should be able to prove the main theorems about these concepts.

16 lectures, 16 tutorials


One formal 2 hour written examination. All questions compulsory.

Full syllabus

1. Vector spaces
Definitions and examples.

2. Linear maps
Definitions and examples.
3. Subspaces
Definitions and examples, (direct) sums and intersections.
4. Independence and spanning sets
Definitions and examples. Bases.
5. Linear maps out of Rn
A linear map Rn→ V is the same as a list of n elements of V.
6. Matrices for linear maps
Definitions, properties, and behaviour under change of basis.
7. Determinant and trace
The definition of determinant and trace for linear operators on a finite dimensional space.
8. Theorems about bases
Invariance of dimension, rank-nullity formula, dim(U+V)+dim(U∩V)=dim(U)+dim(V).
9. Eigenvalues and eigenvectors
A brief account for abstract vector spaces.

Reading list

Type Author(s) Title Library Blackwells Amazon
C Friedberg, Insel, Spence Linear Algebra Blackwells Amazon
C Halmos Finite-dimensional vector spaces Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.