Vector Spaces

Abstracting the Idea of a Vector

Unit 3 treated vectors as arrows and tuples. Linear algebra abstracts away the picture and keeps only the operations. A vector space is any set $V$ of objects (called vectors) equipped with two operations — addition and scalar multiplication — satisfying a short list of axioms. Anything obeying the axioms is a vector space, whatever its elements look like.

The axioms formalize the familiar rules:

• Addition is commutative and associative.
• There is a zero vector $\mathbf{0}$ with $v + \mathbf{0} = v$.
• Every $v$ has an additive inverse $-v$ with $v + (-v) = \mathbf{0}$.
• Scalar multiplication distributes over addition ($c(u + v) = cu + cv$ and $(c + d)v = cv + dv$), is associative ($c(dv) = (cd)v$), and $1v = v$.

The power of abstraction is that many non-arrow objects satisfy these axioms and so are vector spaces:

• $\mathbb{R}^n$ — the familiar tuples.
• Polynomials of degree $\le n$ — you can add them and scale them.
• Matrices of a fixed size — likewise.
• Functions $f : \mathbb{R} \to \mathbb{R}$ — added and scaled pointwise.
• Solutions to a homogeneous linear differential equation.

Because all these share the same axioms, every theorem proved for abstract vector spaces applies to all of them at once. Proving something about "vectors" simultaneously establishes it for tuples, polynomials, matrices, and functions — the efficiency that makes abstraction worthwhile.

Common pitfall: thinking "vector" must mean an arrow or a tuple of numbers. In linear algebra a vector is any element of a set satisfying the vector-space axioms — a polynomial, a matrix, or a function all qualify. What matters is that you can add the objects and scale them consistently; the geometric arrow is just one example, not the definition.