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Created: 2025-09-23
Type: Zettel
aliases:
References:
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tags:
- MATH31AH
---
- **Lecture 1:** Introduction. Injective, surjective, bijective [[functions]]. Read sections 0.4 and 0.6.
- **Lecture 2:** More on bijective functions. Countable and uncountable sets.
- **Lecture 3:** Vectors. Addition, scalar multiplication. Vector subspaces. Linear independence, span, basis, dimension.
- **Lecture 4:** Dot product and cross product. Cauchy-Schwarz and triangle inequalities.
- **Lecture 5:** Systems of linear equations. Matrix-vector product. Matrix products. Transpose.
- **Lecture 6:** Linear transformations, projections, reflections, rotations. Matrix of a linear transformation.
- **Lecture 7:** Composition of linear transformations and matrix multiplication.
- **Lecture 8:** Null space. Examples of null spaces. Null space and uniqueness of solutions of linear systems. Row-reduced echelon form.
- **Lecture 9:** Pivot and free variables. Nullity equals the number of free variables. Basis for null space is obtained from the rref. (In)dependence of columns and vectors in the null space. n+1 vectors in R^n are dependent.
- **Lecture 10:** Column space: Ax=b has solutions only for b in the column space of A. Finding the column space by row-reducing the augumented matrix. Equivalent conditions for columns of A to span, with emphasis on square matrices: rref=I.
- **Lecture 11:** Basis for column space is given by the pivot columns. Rank. Midterm review.
- **Lecture 12:** Rank-nullity theorem. Invertible matrices. Finding the inverse by computing rref[A|I].
- **Lecture 13:** Determinants and invertible matrices. Propreties of determinants stated (no proofs): Expansion along columns and rows. Determinants and row operations. Geometric interpretation of determinants.
- **Lecture 14/15:** More on row operations and determinants. Determinants are alternating and multilinear. Proofs by induction starting from expansion along the first row.
- **Lecture 16:** Uniqueness of the determinant as an alternatating, multilinear, normalized function on the set of n x n matrices. Expansion along rows and columns gives the same answer. Determinant of a product is the product of determinants.
- **Lecture 16:** Determinants and areas/volumes of regions in R^2/R^3.
- **Lecture 17:** Orthogonal complements. Connections between null spaces of a matrix, transpose, column space, column space of the transpose. Examples.
- **Lecture 18:** Matrix of a projection is A(A^TA)^{-1}A^T. Examples. Orthogonal sets. Orthonormal basis of a subspace, for an orthonormal basis the matrix of the projection is AA^T. Orthogonal matrices.
- **Lecture 19:** Another way of computing the projection when an orthonormal basis is given. Finding an orthonormal basis.
- **Lecture 20:** Systems of coordinates. Change of basis matrix. Matrix of a linear transformation with respect to an arbitrary basis. Example: projections onto a plane in R^3.
- **Lecture 21:** Similar matrices. Midterm review.
- **Lecture 22:** Eigenvalues, eigenvectors. Characteristic polynomial is det (\lambda I-A). Eigenvalues are roots of the characteristic polynomial, eigenvectors are found by calculating the null space of \lambda I-A. Diagonalizable matrices are similar to diagonal matrices where the eigenvalues are on the main diagonal.
- **Lecture 23:** Examples of non-diagonalizable matrices. Distinct eigenvalues implies diagonalizable. Characteristic polynomial for 2 x 2 matrix is \lambda^2- Trace \lambda + det. Sum of eigenvalues is the trace, product of eigenvalues is the determinant.
- **Lecture 24:** Complex eigenvalues. A real matrix has eigenvalues in pairs (lambda, lambda conjugate). Symmetric matrices have real eigenvalues. Real symmetric matrices are diagonalizable.
- **Lecture 25:** Examples of orthonormal eigenbasis for symmetric matrices. Quadratic forms. Symmetric matrices associated to quadratic forms. Positive definite, negative definite, semidefinite, indefinite forms.
- **Lecture 25:** Quadratic forms continued: determining the definiteness of the form from the eigenvalues. The cases of 2 x 2 matrices can be read off from the trace and determinant. Examples. What's ahead: abstract vector spaces, abstract linear transformations, etc.
