What you’ll learn
How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.
Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.
Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R2 and R3.
How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.
Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.
Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).
How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.
Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.
Work with various geometrical transformations in R2 and R3, be able to compute their matrices and explain how these transformations work.
Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.
Transform any given basis for a subspace of Rn into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.
Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.
Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.
Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.
Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.
Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).
Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.
Work with various vector spaces, for example with Rn, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.
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Requirements
Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
High-school and college mathematics (mainly arithmetic, some trigonometry, polynomials)
Some basic calculus (used in some examples)
Basic knowledge of complex numbers (used in an example)
Description
Linear Algebra and Geometry 2
Much more about matrices; abstract vector spaces and their bases
Chapter 1 Abstract vector spaces and related stuff
S1. Introduction to the course
S2. Real vector spaces and their subspaces
You will learn: the definition of vector spaces and the way of reasoning around the axioms; determine whether a subset of a vector space is a subspace or not.
S3. Linear combinations and linear independence
You will learn: the concept of linear combination and span, linearly dependent and independent sets; apply Gaussian elimination for determining whether a set is linearly independent; geometrical interpretation of linear dependence and linear independence.
S4. Coordinates, basis, and dimension
You will learn: about the concept of basis for a vector space, the coordinates w.r.t.\ a given basis, and the dimension of a vector space; you will learn how to apply the determinant test for determining whether a set of n vectors is a basis of Rn.
S5. Change of basis
You will learn: how to recalculate coordinates between bases by solving systems of linear equations, by using transition matrices, and by using Gaussian elimination; the geometry behind different coordinate systems.
S6. Row space, column space, and nullspace of a matrix
You will learn: concepts of row and column space, and the nullspace for a matrix; find bases for span of several vectors in Rn with different conditions for the basis.
S7. Rank, nullity, and four fundamental matrix spaces
You will learn: determine the rank and the nullity for a matrix; find orthogonal complement to a given subspace; four fundamental matrix spaces and the relationship between them.
Chapter 2 Linear transformations
S8. Matrix transformations from Rn to Rm
You will learn: about matrix transformations: understand the way of identifying linear transformations with matrices (produce the standard matrix for a given transformation, and produce the transformation for a given matrix); concepts: kernel, image and inverse operators; understand the link between them and nullspace, column space and inverse matrix.
S9. Geometry of matrix transformations on R2 and R3
You will learn: about transformations such as rotations, symmetries, projections and their matrices; you will learn how to illustrate the actions of linear transformations in the plane.
S10. Properties of matrix transformations
You will learn: what happens with subspaces and affine spaces (points, lines and planes) under linear transformations; what happens with the area and volume; composition of linear transformations as matrix multiplication.
S11. General linear transformations in different bases
You will learn: solving problems involving linear transformations between two vector spaces; work with linear transformations in different bases.
Chapter 3 Orthogonality
S12. Gram-Schmidt Process
You will learn: about orthonormal bases and their superiority above the other bases; about orthogonal projections on subspaces to Rn; produce orthonormal bases for given subspaces of Rn with help of Gram-Schmidt process.
S13. Orthogonal matrices
You will learn: definition and properties of orthonormal matrices; their geometrical interpretation.
Chapter 4 Intro to eigendecomposition of matrices
S14. Eigenvalues and eigenvectors
You will learn: compute eigenvalues and eigenvectors for square matrices with real entries; geometric interpretation of eigenvectors and eigenspaces.
S15. Diagonalization
You will learn: to determine whether a given matrix is diagonalizable or not; diagonalize matrices and apply the diagonalization for problem solving (the powers of matrices).
S16. Wrap-up Linear Algebra and Geometry 2
You will learn: about the content of the third course.
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