###### Learning Outcomes:

After successful completion of this module, a student should be able to:

- Define and discuss the main concepts related to vector spaces and linear transformations.

- Construct examples of vector spaces, subspaces and linear transformations.

- Determine if a given set of vectors is linearly independent, a spanning set, a basis.

- Determine if a subset of a vector space is a subspace.

- Determine bases of the row space, column space and null space of a given matrix.

- Determine the rank and nullity of a given matrix.

- Compute the characteristic polynomial, eigenvalues and eigenvectors of a given matrix.

- Determine whether or not a matrix can be diagonalized.

- Compute with coordinatizations of vectors after a choice of basis.

- Compute the change of basis matrix for given bases of a vector space.

- Compute the matrix of a linear map with respect to a choice of bases.

- Determine kernel and image of a linear map.

- Determine if a linear map is injective, surjective, bijective.

- Determine rank and nullity of a linear map.

###### Indicative Module Content:

Brief revision of Gaussian elimination and Gauss-Jordan elimination, vector spaces over a field, subspaces, spanning sets, linear independence, bases, dimension, coordinate spaces, matrix techniques, row space, column space, null space, rank and nullity of a matrix, eigenvalues, eigenvectors, diagonalization, revision of functions on sets and their properties, linear transformations, kernel and image, isomorphisms, matrix of a linear transformation, vector space of linear transformations, change of bases for matrices of linear transformations, rank and nullity of a linear transformation, quotient spaces, dimension formula, first isomorphism theorem, rank-nullity theorem.