VECTOR SPACES

The playground of linear algebra. Understanding dimension, span, and the fundamental blueprint of mathematical reality.

01 // The Span

Linear Combinations

The Span of a set of vectors is every possible coordinate you can reach by stretching and combining them. If you have two vectors pointing in different directions, you can reach the entire infinite 2D plane just by adjusting their scalar multipliers ( $c_1$ and $c_2$ ).

Space extends infinitely.

Must contain Origin

(0,0)

Linear Combinator

The Equation

c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 = \mathbf{w}

Scalar (c₁)1

Stretches

\vec{v}_1

: [2, 1]

Scalar (c₂)1

Stretches

\vec{v}_2

: [-1, 2]

Target Reached

\vec{w}

[1, 3]

Basis

\vec{v}_1

Basis

\vec{v}_2

02 // The Blueprint

Minimum Set

The Basis

The most efficient set of vectors needed to build the space. There can be absolutely no redundancies. They must be Linearly Independent, meaning no vector in the basis can be built by combining the others.

c_1\vec{v}_1 + c_2\vec{v}_2 = \vec{0}

(Only if c=0)

The Count

Dimension

Simply the number of vectors in the Basis. It tells you exactly how many degrees of freedom you have in your space.

1 Vector

\mathbb{R}^1

(Line)

2 Vectors

\mathbb{R}^2

(Plane)

3 Vectors

\mathbb{R}^3

(Space)

03 // The Russian Doll

Subspaces

Spaces Inside Spaces

A line passing perfectly through the origin inside a 3D room is a valid Subspace ( $\mathbb{R}^1$ existing inside $\mathbb{R}^3$ ). Every Matrix creates two fundamental, hidden subspaces:

Col(A) Column Space

The span of all the columns. The reachable outputs.

Nul(A) Null Space

Every input vector that gets crushed to

\vec{0}

\mathbb{R}^3 \supset \mathbb{R}^2 \supset \mathbb{R}^1

Dimensions Verified

You are ready to discover the invariant vectors of Eigen Theory.

Next: Eigen Theory