Linear Algebra: Vectors, Span & Linear Combinations

The Fundamental Object: Vectors

If you had to choose one object to represent all of linear algebra, it would be the **vector**. Everything else—matrices, transformations, eigenvalues—revolves around understanding what vectors are and how they behave.

But what *is* a vector? Ask a physicist, and they'll describe it as an arrow in space with magnitude and direction. Ask a computer scientist, and they'll say it's an ordered list of numbers. Ask a mathematician, and they'll tell you it's anything that can be added and scaled.

**All three perspectives are correct.** We'll start with the geometric view—vectors as arrows—because it provides the most intuition.

Vectors in the Coordinate Plane

Consider a vector in 2D space. We represent it as \(\vec{v} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}\), which means "3 units in the x-direction, 2 units in the y-direction."

Key Insight: The vector \(\vec{v} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}\) is actually a linear combination of the **basis vectors**: \(\vec{v} = 3\hat{i} + 2\hat{j}\).

Vector Addition: Tip-to-Tail

When you add two vectors \(\vec{v} + \vec{w}\), you place the tail of \(\vec{w}\) at the tip of \(\vec{v}\). The result is a vector from the origin to the final tip.

Algebraically: \(\begin{bmatrix} 2 \\ 1 \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \end{bmatrix}\)

Scalar Multiplication: Stretching Space

Multiplying a vector by a scalar (a number) stretches or shrinks it. If the scalar is negative, it also flips direction.

Scalar: 2.0

Linear Combinations: The Heart of It All

A **linear combination** of vectors \(\vec{v}\) and \(\vec{w}\) is any expression of the form:

a\vec{v} + b\vec{w}

Where \(a\) and \(b\) are scalars. This is the most fundamental concept in linear algebra.

                    Example: If \(\vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) and \(\vec{w} =
                    \begin{bmatrix} 3 \\ 1 \end{bmatrix}\), then
                    \(2\vec{v} + 3\vec{w} = \begin{bmatrix} 11 \\ 7 \end{bmatrix}\).
                

Span: All Reachable Points

The **span** of vectors \(\vec{v}\) and \(\vec{w}\) is the **set of all possible linear combinations**:

\text{Span}(\vec{v}, \vec{w}) = \{ a\vec{v} + b\vec{w} \mid a, b \in \mathbb{R} \}

Think of it as: "What are all the points I can reach by scaling and adding these vectors?"

Span of One Vector

The span of a single vector is a **line** through the origin.

\text{Span}(\vec{v}) = \{ a\vec{v} \mid a \in \mathbb{R} \}

Span of Two Vectors

If \(\vec{v}\) and \(\vec{w}\) are **not collinear**, their span is the **entire 2D plane**.

Span of Three Vectors

If you have three vectors in 2D, one is always **redundant**. The span is still the plane.

Linear Independence

Vectors are **linearly independent** if none of them can be written as a linear combination of the others. In other words, no vector is "redundant."

Formal Definition: Vectors \(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\) are linearly
                    independent if:
                    
                        $$ c_1\vec{v}_1 + c_2\vec{v}_2 + \cdots + c_n\vec{v}_n = \vec{0} \implies c_1 = c_2 = \cdots =
                        c_n = 0 $$
                    

Basis: The Coordinate System

A **basis** for a vector space is a set of linearly independent vectors that span the space. In \(\mathbb{R}^2\), the most common basis is:

\hat{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \hat{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}

But this is just **one** choice. Any two non-collinear vectors form a basis for \(\mathbb{R}^2\).

Why it matters: The choice of basis determines your **coordinate system**. Matrix transformations change these basis vectors, which changes how we measure everything.

The Big Picture

Linear algebra is fundamentally about **what you can reach** using linear combinations. Every concept builds on this idea:

Span: What can you reach with a set of vectors?
Basis: What's the minimal set you need to reach everything?
Linear transformations: How do operations preserve linear combinations?
Matrices: How do we represent transformations numerically?

Understanding these geometric foundations will make matrix multiplication, determinants, eigenvalues, and everything else click into place.