||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: scalings, projections, reflections, rotations, shears. Composition of linear transformations and matrix product. Inverse linear transformation and invertible matrices.
||Subspaces of Rn. Linear combinations of vectors. Span of vectors. Linear dependence and independence. Basis. Coordinates. Dimension.
||Kernel and image of a linear transformation. Kernel- Image (Rank-Nullity) theorem.||3.3 |
|| Midterm I on 3/9;
Kernel- Image (Rank-Nullity) theorem. ||3.3
Linear transformations and their matrices. Change of a basis. ||3.4
|| Linear spaces. Linear transformations and their matrices. Isomorphisms. ||4.1-4.2
|| Inner product spaces. Euclidean space Rn. Orthogonality.
|| Orthogonal projections.
Orthogonal complement. Cauchy-Schwarz inequality, Pythagorean Theorem.
Gram-Schmidt orthogonalization. General inner product spaces. || 5.1 -5.2, some of 5.5
|| Midterm II on Apr 20; Determinants.
||Determinants and their geometrical interpretation. Properties of determinants.
|| Eigenvalues and eigenvectors. Eigenspaces. Characteristic equation. Eigenbasis. Diagonalization.
|Friday 5/15 11:00-1:30