here you get to learn from vectors to equation systems to eigen value and eigenvectors . you get to analyze and learn how to work on it as well in the future as well. the perspective is the main key for upgradation in the linear algebra course for the college level all the way.

Operations on one matrix, including solving linear systems, and Gauss-Jordan elimination

Operations on two matrices, including matrix multiplication and elimination matrices

Matrices as vectors, including linear combinations and span, linear independence, and subspaces

Dot products and cross products, including the Cauchy-Schwarz and vector triangle inequalities

Matrix-vector products, including the null and column spaces, and solving Ax=b

Transformations, including linear transformations, projections, and composition of transformations

Inverses, including invertible and singular matrices, and solving systems with inverse matrices

Determinants, including upper and lower triangular matrices, and Cramer's rule

Transposes, including their determinants, and the null (left null) and column (row) spaces of the transpose

Orthogonality and change of basis, including orthogonal complements, projections onto a subspace, least squares, and changing the basis

Orthonormal bases and Gram-Schmidt, including definition of the orthonormal basis, and converting to an orthonormal basis with the Gram-Schmidt process

Eigenvalues and Eigenvectors, including finding eigenvalues and their associate eigenvectors and eigenspaces, and eigen in three dimensions