Chapter 3 Linear algebra

This section is about vector spaces and linear maps. We introduce the idea of a vector space, generalising sets of column vectors, and a linear map: a function between two vector spaces which preserves their operations of addition and scalar multiplication.

Learning objectives for this section:

• Know the definitions of vector space, subspace, linear combination, span, linear independence, spanning set, basis, dimension, linear map, kernel, image, rank, nullity, matrix of a linear map, eigenvalue, eigenvector, diagonalizable.
• Verify whether a given subset is a subspace.
• Verify whether a sequence of vectors is linearly independent, a spanning set, or a basis.
• Compute bases for and dimensions of subspaces.
• Understand the relationship between the dimension of a vector space and its subspaces.
• Recognise linear maps, find their kernels and images and rank and nullity.
• Find the matrix of a linear map.
• State the relationship between matrices of linear maps with respect to different bases.
• Find eigenvalues and eigenvectors of linear maps in simple cases.
• Determine whether a linear map is diagonalizable.
• State the relationship between diagonalizability and eigenvectors.