Linear Algebra

What Is a Vector?

A vector is a short list of numbers that tells you both a direction and a size.
Imagine an arrow drawn on a piece of paper: the arrow points somewhere and has a certain length.
In math we write that arrow as a pair ([x, y]) for a flat (2‑D) picture, or as a triple ([x, y, z]) for a 3‑D picture.

• Adding Vectors – Put the tail of one arrow at the tip of another. The new arrow goes from the start of the first to the end of the second.
• Scaling A Vector – Multiply every number by the same amount. The arrow gets longer or shorter, but it still points the same way.

Vectors let us describe motion, forces, and many other things that have both size and direction.

Matrices and How They Work

A matrix is a rectangular table of numbers, like a tiny spreadsheet.
Rows go left‑to‑right, columns go top‑to‑bottom.

Adding Matrices

You can add two matrices only when they have the same shape.
Just add the numbers that sit in the same row and column.

Multiplying a Matrix by a Vector

When you multiply a matrix by a vector, you change the vector’s direction and length.
The rule is: each entry of the result comes from one row of the matrix multiplied by the whole vector, then added together.

Matrix‑by‑matrix Multiplication

If you have two matrices, the entry in row i and column j of the product is the sum of the products of matching numbers from row i of the first matrix and column j of the second.
This operation lets us combine several transformations (like rotating then stretching) into one step.

Why Linear Algebra Is Useful

Linear algebra is the language for many real‑world problems.

• Solving Many Equations At Once – Systems of linear equations appear in physics, engineering, and economics.
• Modeling Networks – Traffic flow, social connections, and computer networks can be written as matrices.
• Computer Graphics – Video games and movies use matrices to move, rotate, and scale 3‑D objects on screen.
• Future Topics – The same ideas appear in machine learning, data science, and even quantum physics.

By learning vectors and matrices now, you build a toolbox that will help you understand and create technology, solve puzzles, and explore how the world works.