Vector space
A set with vector addition and scalar multiplication over a field F satisfying the usual distributive, associative, identity, and inverse axioms.
Find a concept
REFERENCE SHEET
Vector spaces & linear maps
Separate coordinate-free objects from the matrices used to describe them.
Before this Sets & maps
dim V = dim ker T + dim im T
Definitions
Basis
A linearly independent spanning set. In finite dimension, each vector has a unique linear combination in that basis.
Kernel and image
For a linear map T: V→W, ker T={v:T(v)=0} and im T={T(v):v∈V}; both are subspaces.
Key theorems
Rank–nullity
For T: V→W with V finite-dimensional, dim V=dim ker T+dim im T.
Basis extension
Every linearly independent set in a finite-dimensional vector space extends to a basis.
Notable applications
Constraints
Null spaces describe motions or parameter changes that preserve linear constraints.
Representations
A group can act by invertible linear maps on a vector space, turning symmetry into linear algebra.
Work it out
Try before revealing01 / Compute
For T(x,y,z)=(x+y,y+z), find a basis of ker T and the rank.
Hint
Solve x+y=0 and y+z=0.
Solution
The kernel is span{(−1,1,−1)}, so nullity=1 and rank=3−1=2.
02 / Prove
Show that an injective linear map between two finite-dimensional spaces of the same dimension is surjective.
Hint
Injectivity means the kernel is zero.
Solution
Rank–nullity gives rank T=dim V=dim W, so the image subspace is all of W.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: