Definitions:

• null space of T: null T = {v ∈ V: Tv = 0} (stuff in the original vector space that’s mapped to 0 by T)
• range of T: range T = {Tv: v ∈ V} (all the resulting vectors after the mapping T)
• A function T: V -> W is injective if Tu = Tv implies u = v.
• A function T: V -> W is surjective if its range equals W.
• An isomorphism is an invertible (injective + surjective) linear map.
• Two vector spaces are isomorphic if there is an isomorphism from one vector space onto the other one.

Facts:

• A linear map is injective iff null T = {0}. [Proof Idea: if v ∈ null T then T(v) = 0 = T(0) => v = 0 so null T ⊆ {0}; other direction trivial.]

• Rank-Nullity: if T ∈ L(V, W) then dim V = dim null T + dim range T

• A map to a smaller dimensional space is not injective. (“smaller” as measured by dimension) [Proof Idea: using Rank-Nullity]

• A map to a larger dimensional space is not surjective.

• If T is invertible, it’s inverse is unique.

• Two finite-dimensional vector spaces over F are isomorphic if and only if they have the same dimension. [Proof Idea: consider T: V -> W; =>: using Rank-Nullity; <=: denote basis of V, W. Apply T to the linear combination of the basis of V, which results in the same linear combination of the basis of W, we can show T is surjective and injective, and is thus invertible.]

• L(V, W) and F^m,n are isomorphic

• dim F^m,n = mn [Proof Idea: m-by-n matrices that have 0 in all entries except for a 1 in one entry form a basis of F^m,n. How many of them are there?]

• Based on the previous two facts, dim L(V, W) = (dim V)(dim W)