The goals for 18.06 are *using matrices and also understanding them*.
Here are key computations and some of the ideas behind them:

1. Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A = LU)

2. Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special solutions to Ax = 0 from row reduced R)

3. Basis and dimension (bases for the four fundamental subspaces)

4. Least squares solutions (closest line by understanding projections)

5. Orthogonalization by Gram-Schmidt (factorization into A = QR)

6. Properties of determinants (leading to the cofactor formula and the sum over all n! permutations, applications to inv(A) and volume)

7. Eigenvalues and eigenvectors (diagonalizing A, computing powers A^k and matrix exponentials to solve difference and differential equations)

8. Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for x'Ax > 0, applications)

9. Linear transformations and change of basis (connected to the Singular Value Decomposition -- orthonormal bases that diagonalize A)

10. Linear algebra in engineering (graphs and networks, Markov matrices, Fourier matrix, Fast Fourier Transform, linear programming)