LinearSubspace.md (917B)
1 # Linear Subspace (or simply subspace) 2 3 Khan 4 5 **Definition:** A linear subspace is a subset (inclusive of the subset being the entire set) of a space of equal or greater cardinality where the linear subspace contains the zero vector. 6 7 Things like a plane that passes through the origin in R^n, a line that passes through the origin in R^n, or R^n itself are all specific linear subspaces (or just subspaces for short). 8 9 Additionally, we have closure under addition meaning any two vectors added together in the set are still contained within the subspace and the superspace. 10 11 The final thing is it is closed under multiplication. As such, multiplying any vector in the set by any scalar means the result is also in the subspace. 12 13 ### Summary 14 15 Three rules neccesary and sufficient for subspace definition: 16 17 1. Closed under multiplication (scalar) 18 2. Closed under addition (with other elements) 19 3. Contains zero vector