||Linear systems and their geometric interpretation. Matrices and vectors. The matrix form of a linear system. Gauss-Jordan elimination.
||Matrix vocabulary. Operations on matrices. Space Rn. Rank of a matrix.
Number of solutions of a linear system.
Linear transformations from Rm to Rn. Matrix of a linear transformation.
Linear transformations in a plane and how to visualize them.
The notion of inverse matrix.
|9/7|| More on geometry of linear transformations and their matrices. Geometric meaning of dot product.
Projections and reflections in the plane and the 3-space; scalings, rotations, and shears.
Appendix A (no cross product)
|9/14 || Matrix multiplication and its meaning.
Inverse transformations and their matrices.
Kernel and Image of a linear transformation.
Subspaces, span of vectors, linear (in)dependence. Basis.
Finding Image (as span of column vectors)
and kernel (solve the system!) of a linear transformation.
Dimension of a subspace.
Geometric interpretation of rank (as dimension of image).
Rank-nullity theorem and how to use it.
Coordinates of a vector with respect to a basis.
Finding a matrix of linear transformation in a new basis.
More on invertible matrices.
Further applications/discussion of Chapter 3.
Begin Chapter 4: more general linear spaces are useful!
Exam on 10/9 covering Chapters 1-3.
Chapter 4: more general linear spaces.
4.1 Linear spaces and subspaces, span, basis
4.2 Linear transformations, isomorphisms, coordinates.
Linear transformations in arbitrary vector spaces,
isomorphisms, the matrix of a transformation in a particular basis.
| 4.2, 4.3
Orthogonality, orthogonal projections, orthonormal basis.
Gram-Schmidt process. Orthogonal matrices.
| 5.1, 5.2, 5.3