Vectors and Vector Spaces

What is a Vector?

A vector is an object with magnitude and direction, like velocity or force. In $n$ dimensions, it is an $n$ -tuple. A vector space is any set of objects that can be added and scaled by numbers (scalars), obeying natural rules. This abstraction applies to arrows, polynomials, matrices, and functions.

Two core operations define the space:

Operation	Rule	Geometric meaning
Vector addition	$(a_1, b_1) + (a_2, b_2) = (a_1+a_2, b_1+b_2)$	Tip-to-tail placement
Scalar multiplication	$c(v_1, v_2) = (cv_1, cv_2)$	Stretch, shrink, or reverse

These operations obey vector-space axioms: commutative and associative addition, a zero vector $\mathbf{0}$ that changes nothing, additive inverses, and distributive laws.

Length and Pitfalls

The magnitude (norm) of a vector measures its length. For a 2D vector, the Pythagorean norm formula is:

$\|v\| = \sqrt{v_1^2 + v_2^2}$

A unit vector has norm 1 and captures pure direction. In computing, vectors serve as graphics coordinates, machine learning feature vectors, and language model embeddings.

Concept	Correct Meaning	Common Mistake
Magnitude	Single length $\sqrt{v_1^2 + v_2^2}$	Treating components $v_1, v_2$ as the length
Addition	Componentwise addition	Adding magnitudes directly ( $\\|u+v\\| \neq \\|u\\| + \\|v\\|$ )

Pitfall: Do not treat vectors as mere lists of numbers. A vector is defined by how it adds and scales, and the length of a vector sum is generally not the sum of their individual lengths.

Vectors and Vector Spaces

What is a Vector?

Length and Pitfalls

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Algebra