Weeks 1-3:
Vector space definition,
Linearly independent vectors, spans
Basis of a vector space
Sifting, the existence of basis.
Examples

Weeks 4-6:
Linear transformations,
Kernel, Image
The dimension formula
Relation between matrices and linear transformations.
Change of bases formula

week 7:
Eucledian spaces
Orthonormal basis
Orthogonal transformations

week 8:
Determinant and its properties

week 9-10:
Linear operators
Eigenvalues and eigenvectors
Characteristic polynomial