# ## Videos and Tutorials: Linear Maps (Matrices)

In this part, I give examples (without a formal definition) of linear functions between vector spaces and show that a linear function between finite dimensional vector spaces can be represented by a matrix.

In this part, I introduce the formal definitions of linear maps, injective, surjective and bijective maps. I introduce the concept of a kernel of a map and how it relates to the invertibility of a matrix. I also talk about the composition of linear operators (multiplication of matrices)

In this part, I introduce the Kronecker Delta, the Alternator and the Einstein Summation Convention with some examples. Then, I touch on the function of the “Determinant” of a matrix as a measure of the invertibility of a matrix. Finally, I present the idea of a matrix transpose, the inverse and the eigenvalues and eigenvectors.

In this part, I start by giving examples of rotation matrices. Then, I formally introduce orthogonal matrices in three dimensional Euclidean vector spaces and then talk about the difference between rotation and reflection matrices. Then, I discuss the difference between the usage of orthogonal matrices for rotation of objects and for change of basis. Finally, I introduce the concept of invariants and matrix invariants in 3 dimensional Euclidean vector spaces.

In this video, I present the symmetric matrices and their diagonalization procedure (spectral decomposition of symmetric matrices) along with positive definite and semi-positive definite symmetric matrices.

In this video, I present how to manipulate matrices in Mathematica. In particular, how to construct and use rotation matrices, how to find the eigenvalues and eigenvectors of symmetric matrices and how to extract individual eigenvalues and eigenvectors.