Linear Algebra and Geometry 2

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 R^n.

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 R^n 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 R^n to R^m

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 R^2 and R^3

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 R^n; produce orthonormal bases for given subspaces of R^n 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.

S17. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.

Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

A detailed description of the content of the course, with all the 214 videos and their titles, and with the texts of all the 153 problems solved during this course, is presented in the resource file

“001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf”

under video 1 (“Introduction to the course”). This content is also presented in video 1.