Linear Algebra Matrix

Video Resources#

• Vectors | Chapter 1, Essence of linear algebra
• Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Geometric Perspective#

A vector is an arrow with magnitude and direction. In 2D you can think of it as moving across the x and y axes.

Algebraic Perspective#

A vector is also an ordered list of numbers:

• 2D: (x, y)
• 3D: (x, y, z)
• nD: (v1, v2, ..., vn)

Core Operations#

Addition#

Add vectors component-wise:

(1, 2, 3) + (4, 5, 6) = (5, 7, 9)

Scalar Multiplication#

Scale each component by a constant:

c * (x, y, z) = (cx, cy, cz)

Dot Product#

The dot product measures alignment and is used heavily in similarity and distance-based models.

Cross Product#

The cross product creates a vector perpendicular to two 3D vectors.