BitWithBite
Linear Algebra · Quick Reference
Vector Spaces Cheat Sheet
In one line: a vector space is a collection of objects (vectors) that can be added together and scaled, following a consistent set of rules — an abstraction that unifies everything from arrows in space to functions and polynomials.
Key Ideas
1Vector Space. A set of vectors that is closed under addition and scalar multiplication, satisfying rules like associativity, commutativity, and the existence of a zero vector.
2Subspace. A subset of a vector space that is itself a vector space — it must contain the zero vector and be closed under addition and scalar multiplication.
3Linear Combination. Any expression formed by scaling and adding vectors together, like 3*v1 + 2*v2 - v3.
4Span. The set of every possible linear combination of a group of vectors — essentially everything you can 'reach' using just those vectors.
5Linear Independence. A set of vectors is linearly independent if no vector in the set can be written as a combination of the others (the only way to get the zero vector is if every coefficient is zero).
Worked Examples
Is the set {(1,0), (0,1)} linearly independent in 2D space?
Yes, they are linearly independent
Is the set {(1,2), (2,4)} linearly independent?
No, they are linearly dependent
What is the dimension of standard 2D space (the plane)?
Dimension = 2
Practice Yourself
Is {(1,0,0),(0,1,0),(0,0,1)} linearly independent?
Yes
Is {(1,1),(2,2)} linearly independent?
No, the second is a multiple of the first
What is the dimension of standard 3D space?
3
What must a subspace always contain?
The zero vector