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Introduction

Hey there, I’m Vikram Kommera. I have been taking notes using Obsidian so far this quarter. I have typed my notes in LaTeX using the LaTeX Suite plugin for Obsidian, which allows me to use shortcuts while typing to keep up during lecture.

I have taken notes on most proofs and examples during lecture. There may be typos and conceptual mistakes, but I have tried my best to keep this accurate. I hope you can find this useful to study for our final.

Lecture Summaries

This only roughly maps to the lectures we had in class. I took this from out canvas page and linked different notes as a I found them. There are more notes that linked here. See the sidebar for all notes.

• 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.