Vectors and Vector Spaces
What Are Vectors?
Vectors are mathematical objects that have both direction and magnitude. You can imagine a vector as an arrow pointing from one place to another in space. In math, we often write vectors as ordered lists of numbers, like \( \mathbf{v} = (2, 3) \) in two dimensions.
Exploring Vector Spaces
A vector space is a collection of vectors that you can add together and multiply by numbers (called scalars) to get new vectors. For a set to be a vector space, it must follow certain rules, such as:
- Adding any two vectors in the space gives another vector in the space.
- Multiplying any vector by a scalar (like 2 or -1) keeps it in the space.
Why Do We Care About Vectors?
Vectors are everywhere! They describe movement, forces, and even data in computer science.
Real-World Connections
- Physics uses vectors to describe velocity and force.
- Computer graphics use vectors to move and rotate images.
Operations on Vectors
- Addition: \( (1,2) + (3,4) = (4,6) \)
- Scalar Multiplication: \( 2 \times (1,2) = (2,4) \)
Working with vectors is like working with arrows—combine them, stretch them, or shrink them!
Examples
Adding the vectors (1, 2) and (3, 4) gives (4, 6).
Doubling the vector (2, -5) makes (4, -10).
In a Nutshell
Vectors have magnitude and direction; vector spaces are collections of vectors that can be added and scaled.