Linear Transformations, Rank, and Nullity
Key definitions, examples, and proof ideas behind linear maps and the rank–nullity theorem.
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Questions Covered in This Set
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Define a linear transformation T: V → W.
A function satisfying T(u+v)=T(u)+T(v) and T(cv)=cT(v) for all u,v ∈ V and scalars c. Consequences: T(0)=0 and T(Σcᵢvᵢ)=ΣcᵢT(vᵢ).
Why is a linear map 'determined by a basis'?
Because T respects all linear combinations: choose images T(e₁),…,T(eₙ) of a basis freely, and exactly one linear map does that. Stacking those image coordinate columns gives the matrix [T], with [T][v]=[T(v)].
Define ker T and im T, and their dimensions.
ker T = {v ∈ V : T(v)=0} ⊆ V, im T = {T(v) : v ∈ V} ⊆ W. nullity(T)=dim ker T, rank(T)=dim im T. Both are subspaces (one-line check with T(u+cv)=T(u)+cT(v)).
Prove: T injective ⟺ ker T = {0}.
(⇒) T(0)=0, so injectivity forces 0 to be the only kernel element. (⇐) If T(u)=T(v), then T(u−v)=0, so u−v ∈ ker T = {0}, hence u=v. Linearity lets you test injectivity at a single point.
State the Rank–Nullity Theorem.
If V is finite-dimensional and T: V → W is linear, then dim ker T + dim im T = dim V.
Sketch the proof of rank–nullity.
Take a basis {u₁,…,u_k} of ker T, extend to a basis {u₁,…,u_k,w₁,…,w_{n−k}} of V. Show {T(w₁),…,T(w_{n−k})} spans im T (the uᵢ terms die) and is independent (a dependence puts Σb_j w_j in ker T, forcing all b_j = 0). So rank = n − k.
Differentiation D on P₃: matrix, kernel, image.
With basis 1,x,x²,x³: D(1)=0, D(x)=1, D(x²)=2x, D(x³)=3x² — strictly upper triangular with 1,2,3 on the superdiagonal. ker D = constants (nullity 1), im D = P₂ (rank 3), and 1+3=4=dim P₃.
Give the rotation matrix of ℝ² by θ and how it's derived.
Ask where the basis goes: e₁ ↦ (cos θ, sin θ), e₂ ↦ (−sin θ, cos θ). Columns give [[cos θ, −sin θ],[sin θ, cos θ]].
Name three maps that are NOT linear and why.
x ↦ x+1 (moves the origin, T(0)≠0), x ↦ x² (fails additivity/homogeneity), v ↦ ‖v‖ (fails T(cv)=cT(v) for c<0).
Rank and nullity of projection (x,y,z) ↦ (x,y,0).
Kernel is the z-axis (nullity 1), image is the xy-plane (rank 2); 1+2=3=dim ℝ³.
