1. MAT 2401 Linear Algebra
4.1 Vectors in Rn

2. HW
Written HW
Practice HW:7, 9, 15, 21,23, 27, 29, 31

3. Overview: Apples, Triangles and Abstractions
Common Properties

4. Overview: Apples, Triangles and Abstractions
Let us name our collection the BIG RED.

5. Overview: Apples, Triangles and Abstractions
Common Properties

6. Likewise,…
We can classify mathematical objects based on some common properties.
Mathematical
Object A
Mathematical
Object B
Mathematical
Object C
Mathematical
Object D

7. Classifications in Your Discipline?
Chemistry
Engineering
Physics
Business

8. Overview 4.1 Vectors in Rn 4.2 (Abstract) Vector Spaces
Easy to understand mathematical objects and properties.
4.2 (Abstract) Vector Spaces
Based on some of the important properties of real vector spaces, we find other objects that share the same collection of properties.
These objects are all refer as (abstract) vector spaces.

9. Overview 4.3 (skipped since 2015)
4.4, 4.5 (necessary details to move on)
Purposes of Abstraction
If one can prove a result for the abstract vector spaces, the result applies to all member of the collection.
For example, …

10. Overview: Apples, Triangles and Abstractions
If we show that BIG RED ….

11. One Theorem to Rule Them All...
Concrete examples that can be understand at this level is (again) difficult to come by.
In the past, LA was often taught alongside Differential Equations because DE can illustrate the application of vector spaces (and it can be understood at this level).

12. The Journey
From this point on, we will go into a little bit of abstractions.
Keep the real vector spaces in mind as the concrete, tangible example.
Today, the discovery lab will be on …

13. Notations Matrices Calculus Row vector: [1 2 1] Vector in R3: (1,2,1)
If you already known some of these content from calculus III, think of the lab as a review.