GMAT Focus | Quantitative| Master Math Course | 2026 Updated
Nadeem Arain
5.0 (853)·55h 15m · 544 lectures
$3.00$12–$199up to −98%
Loading course
Linear Algebra and Geometry 2
Much more about matrices; abstract vector spaces and their bases; linear transformations animated with MANIM
5.0328 reviews6,182 learners
- All Levels
- 47h 9m video
- 215 lectures
- English
- Captions
- Updated 2026
Taught by Hania Uscka-Wehlou
- 47h 9m of on-demand video
- 215 lectures across 17 sections
- One-off payment — no subscription
- Lifetime access on web and mobile
- Closed captions (English)
Curriculum
17 sections215 lectures47h 9m total
- Introduction to the course14:11
- From abstract to concrete4:05
- From concrete to abstract3:54
- Our prototype12:35
- Formal definition of vector spaces Example 1: Rn18:48
- Vector spaces, Example 2: m × n matrices with real entries16:57
- Vector spaces, Example 3: real-valued functions on some interval18:26
- Vector spaces, Example 4: complex numbers14:40
- Cancellation property6:52
- Two properties of vector spaces; Definition of difference14:35
- Some properties of vector spaces16:28
- What is a subspace20:18
- All the subspaces in R217:38
- All the subspaces in R310:39
- Subspaces, Problem 19:26
- Subspaces, Problem 235:12
- Subspaces, Problem 322:17
- Subspaces, Problem 411:42
- Our unifying example6:51
- Linear combinations in Part 110:50
- Linear combinations, new stuff. Example 16:38
- Linear combinations Example 25:42
- Linear combinations, Problem 115:39
- Linear combinations, Problem 227:48
- What is a span, definition and some examples9:12
- Span, Problem 315:57
- Span, Problem 416:32
- Span, Problem 52:10
- What do we mean by trivial?6:33
- Linear independence and linear dependence14:50
- Geometry of linear independence and linear dependence15:20
- An important remark on linear independence in Rn10:37
- Linearly independent generators, Problem 624:07
- Linear independence in the set of matrices, Problem 716:24
- Linear independence in C^0[−∞, ∞], Problem 810:36
- Vandermonde determinant and polynomials18:41
- Linear independence in C^∞(R), Problem 917:41
- Wronskian and linear independence in C∞(R)9:33
- Linear independence in C^∞(R), Problem 105:47
- Linear independence in C^∞(R), Problem 119:02
- What is a basis and dimension?8:57
- Bases in the 3-space, Problem 140:36
- Bases in the plane and in the 3-space13:11
- Bases in the 3-space, Problem 25:53
- Bases in the 4-space, Problem 319:23
- Bases in the 4-space, Problem 428:21
- Bases in the space of polynomials, Problem 54:36
- Coordinates with respect to a basis12:28
- Coordinates with respect to a basis are unique8:23
- Coordinates in our unifying example4:57
- Dimension of a subspace, Problem 611:02
- Bases in a space of functions, Problem 710:57
- Coordinates in different bases6:37
- It is easy to recalculate from the standard basis6:45
- Transition matrix, a derivation13:36
- Previous example with transition matrix4:59
- Our unifying example9:22
- One more simple example and bases4:58
- Two non-standard bases, Method 18:07
- Two non-standard bases, Method 213:46
- How to recalculate coordinates between two non-standard bases? An algorithm13:04
- Change of basis, Problem 139:40
- Change of basis, Problem 213:42
- Change of basis, Problem 324:14
- Change of basis, Problem 48:33
- Change of basis, Problem 512:04
- Change to an orthonormal basis in R^214:38
- What you are going to learn in this section5:49
- Row space and column space for a matrix9:52
- What are the elementary row operations doing to the row spaces?19:59
- What are the elementary row operations doing to the column spaces?17:11
- Column space, Problem 213:31
- Determining a basis for a span, Problem 314:33
- Determining a basis for a span consisting of a subset of given vectors, Problem416:18
- Determining a basis for a span consisting of a subset of given vectors, Problem517:11
- A tricky one: Let rows become columns, Problem 611:26
- A basis in the space of polynomials, Problem 717:59
- Nullspace for a matrix13:08
- How to find the nullspace, Problem 88:34
- Nullspace, Problem 917:51
- Nullspace, Problem 1024:48
About this course
Linear Algebra and Geometry 2 Much more about matrices; abstract vector spaces and their bases [None of our courses are produced using AI; they are all real-human products.] 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 independen…
Before you start
- 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 that can be omitted without having any negative impact on your studies of Linear Algebra, and revisited after you have studied Calculus: some simple derivatives are needed in Videos 38, 39, 40, 132, 134, 136; continuous functions are mentioned in Videos 16, 35--40)
- Basic knowledge of complex numbers (used in an example in Video 8)
- You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.
Who it’s for
- University and college engineering
Related
More in Teaching & Academics
How to Become a Life Coach - Is Training Needed?
Maia Berens
5.0 (662)·3h 5m · 43 lecturesNot available yetCurso Básico de Russo e Vocabulário Para Turismo
Dimitri Olegovich Bobrovnikov
5.0 (325)·5h 27m · 33 lectures$3.00$12–$199up to −98%GRE - GMAT | Advanced Learners | Math Marathon
Nadeem Arain
5.0 (273)·10h 43m · 194 lectures$3.00$12–$199up to −98%
$3.00$12–$199