Source: Friedberg, Insel & Spence, Linear Algebra 4th Ed.
Tags: linear transformation, null space, kernel, range, image, rank, nullity, dimension theorem, one-to-one, onto, injective, surjective, linear map, linear operator
Difficulty: Intermediate Prerequisites: Chapter 1 (vector spaces, bases, dimension, subspaces, linear independence, span). You should be comfortable with the definition of a vector space over a field F, what a basis is, and how dimension works.
This section introduces linear transformations, the central objects connecting abstract vector spaces to concrete computation. If Chapter 1 built the stage (vector spaces), Chapter 2 puts actors on it (the functions that respect vector space structure). Linear transformations appear everywhere: differentiation and integration in calculus, rotations and reflections in geometry, and matrix multiplication in applied science. Understanding the null space and range of a transformation tells you what information it destroys and what it can produce, which is the key to solving linear equations, determining invertibility, and much more.
A linear transformation is a function between vector spaces that preserves addition and scalar multiplication. Every linear transformation has a null space (what gets sent to zero) and a range (what the transformation can hit). The Dimension Theorem ties these together: nullity + rank = dim(V). A transformation is one-to-one precisely when its null space is trivial, and for equal-dimension spaces, one-to-one and onto are equivalent.
Linear transformation (linear map)
A function T: V → W between vector spaces (over the same field F) satisfying two conditions for all x, y in V and all scalars c in F: (a) T(x + y) = T(x) + T(y), and (b) T(cx) = cT(x). In simple terms, T "respects" the vector space operations. It does not bend addition or scalar multiplication out of shape.
Null space (kernel), N(T)
The set of all vectors x in V such that T(x) = 0. Formally, N(T) = {x in V : T(x) = 0}. Think of it as everything the transformation crushes to zero. It is always a subspace of the domain V.
Range (image), R(T)
The set of all outputs of T, i.e. R(T) = {T(x) : x in V}. This is the collection of vectors in W that T can actually produce. It is always a subspace of the codomain W.
Nullity, nullity(T)
The dimension of the null space N(T). It measures how much information T destroys.
Rank, rank(T)
The dimension of the range R(T). It measures how much of the codomain T can reach.
Dimension Theorem (Rank-Nullity Theorem)
For T: V → W linear with V finite-dimensional: nullity(T) + rank(T) = dim(V). The bigger the null space, the smaller the range, and vice versa. This is one of the most-used results in linear algebra.
Identity transformation, I_V
Defined by I_V(x) = x for all x in V. Its null space is {0} and its range is all of V.
Zero transformation, T_0
Defined by T_0(x) = 0 for all x in V. Its null space is all of V and its range is {0}.
T: V → W is linear if it preserves both addition and scalar multiplication.
The standard shortcut for proving linearity: show T(cx + y) = cT(x) + T(y) for all x, y in V and c in F. This combines both conditions into one check.
Four immediate consequences of linearity:
T(0) = 0 (linear maps always send the zero vector to zero)
T(x − y) = T(x) − T(y)
T preserves arbitrary finite linear combinations: T(sum of a_i x_i) = sum of a_i T(x_i)
If T(0) is not 0, the map cannot be linear (quick disqualifier)
Rotation by angle θ in R²: T_θ(a₁, a₂) = (a₁ cos θ − a₂ sin θ, a₁ sin θ + a₂ cos θ). Rotates every vector counterclockwise by θ.
Reflection about the x-axis in R²: T(a₁, a₂) = (a₁, −a₂). Flips the second coordinate.
Projection onto the x-axis in R²: T(a₁, a₂) = (a₁, 0). Drops the vertical component.
Matrix transpose: T(A) = Aᵗ on M_{m×n}(F).
Differentiation: T: P_n(R) → P_{n−1}(R) defined by T(f(x)) = f'(x). Linear because derivatives distribute over sums and pull out constants.
Integration: T(f) = integral from a to b of f(t) dt. Linear by the same properties of integrals.
Theorem 2.1: If T: V → W is linear, then N(T) is a subspace of V and R(T) is a subspace of W. The proof checks closure under addition and scalar multiplication directly, using linearity of T.
Theorem 2.2: If β = {v₁, …, vₙ} is a basis for V, then R(T) = span({T(v₁), T(v₂), …, T(vₙ)}). You only need to know what T does to a basis to determine the entire range. To find a basis for R(T), apply T to each basis vector and then reduce the resulting set to a linearly independent one.
Theorem 2.3: nullity(T) + rank(T) = dim(V).
Proof sketch: start with a basis {v₁, …, v_k} for N(T), extend it to a basis {v₁, …, v_k, v_{k+1}, …, vₙ} for V. Then {T(v_{k+1}), …, T(vₙ)} is a basis for R(T). So rank(T) = n − k, which gives the result.
This is the single most important bookkeeping tool for linear transformations. If you know two of the three quantities (nullity, rank, dim V), you can find the third.
Theorem 2.4: T is one-to-one (injective) if and only if N(T) = {0}. To check injectivity, you only need to check whether the null space is trivial.
Theorem 2.5: When dim(V) = dim(W) (both finite), three conditions are equivalent: (a) T is one-to-one, (b) T is onto (surjective), (c) rank(T) = dim(V). For equal-dimension spaces, you get injectivity and surjectivity for free once you establish either one.
This equivalence fails for infinite-dimensional spaces. The left shift operator (drop the first term of a sequence) is onto but not one-to-one; the right shift (prepend a zero) is one-to-one but not onto.
Theorem 2.6: Given a basis {v₁, …, vₙ} for V and any vectors w₁, …, wₙ in W, there exists exactly one linear transformation T: V → W with T(v_i) = w_i for each i. This means specifying T on a basis completely determines T everywhere.
Linearity shortcut: T(cx + y) = cT(x) + T(y) for all x, y in V, c in F
Dimension Theorem: nullity(T) + rank(T) = dim(V)
Rotation formula in R²: T_θ(a₁, a₂) = (a₁ cos θ − a₂ sin θ, a₁ sin θ + a₂ cos θ)
Rotations, reflections, and projections are exactly the transformations used in computer graphics to manipulate objects on screen. Every time a game engine rotates a 3D model or a CAD program mirrors a shape, it is applying a linear transformation. The null space tells an engineer which inputs a sensor cannot distinguish (they all produce zero output), while the range tells them the set of possible readings.
"If T(x + y) = T(x) + T(y), then T must be linear." Over the reals, additivity alone does not guarantee T(cx) = cT(x). Over the rationals it does, but the general case requires checking both conditions (or using the combined shortcut).
"A linear transformation can send 0 to something other than 0." It cannot. T(0) = 0 is always true for a linear map. If a function fails this, it is not linear.
"One-to-one and onto are always equivalent for linear maps." They are equivalent only when V and W have the same finite dimension. If dim(V) < dim(W), T cannot be onto; if dim(V) > dim(W), T cannot be one-to-one.
"To find the range, I need to compute T(x) for every x." You only need T applied to a basis. Theorem 2.2 says the range is the span of those images.
⚠️ The Dimension Theorem is tested constantly. Expect questions that give you two of the three quantities and ask for the third.
⚠️ "Show T is linear" is a standard exam prompt. Use the combined condition T(cx + y) = cT(x) + T(y) and show both sides are equal.
⚠️ "Determine whether T is one-to-one / onto" is typically answered by computing the null space or the rank, then applying Theorems 2.4 and 2.5.
⚠️ Theorem 2.6 (uniqueness from action on a basis) is the theoretical backbone for many later results. Understand its proof.
1. True or false: If T is linear, then T(0) could equal any vector in W.
A: False. T(0) = 0 always.
2. Fill in the blank: nullity(T) + ______ = dim(V).
A: rank(T)
3. True or false: If T: R⁵ → R³ is linear, T could be one-to-one.
A: False. dim(V) > dim(W), so T cannot be injective.
4. True or false: If N(T) = {0} and dim(V) = dim(W), then T is onto.
A: True, by Theorem 2.5.
5. Fill in the blank: R(T) = span({T(v₁), …, T(vₙ)}) whenever {v₁, …, vₙ} is a ______ for V.
A: basis
Q: Let T: R³ → R² be defined by T(a₁, a₂, a₃) = (a₁ − a₂, 2a₃). Find N(T), R(T), the nullity, and the rank. Is T one-to-one? Is T onto?
A: N(T) = {(a, a, 0) : a in R}, so nullity = 1. R(T) = R² (you can hit any (b₁, b₂) by choosing a₁ − a₂ = b₁ and a₃ = b₂/2), so rank = 2. Check: 1 + 2 = 3 = dim(R³). T is not one-to-one (nullity > 0). T is onto (rank = dim(R²) = 2).
Q: Define T: P₂(R) → P₃(R) by T(f(x)) = 2f'(x) + integral from 0 to x of 3f(t) dt. Determine whether T is one-to-one and whether T is onto.
A: Compute T on the basis {1, x, x²}: T(1) = 3x, T(x) = 2 + (3/2)x², T(x²) = 4x + x³. These three outputs are linearly independent, so rank(T) = 3. Since dim(P₂(R)) = 3, nullity = 0, so T is one-to-one. Since dim(P₃(R)) = 4 > 3 = rank(T), T is not onto.
Q: Give an example of a function T: R² → R² that satisfies T(x + y) = T(x) + T(y) for all x, y but is NOT linear (over the reals).
A: Over R (not Q), construct a pathological additive function using a Hamel basis for R over Q. These are non-measurable and cannot be written with a simple formula, but they exist by the Axiom of Choice. Any such function satisfies additivity but not T(cx) = cT(x) for irrational c.
Q: Suppose T: V → V is linear, dim(V) = 4, and rank(T) = 2. What is the nullity? Could T be one-to-one?
A: Nullity = 4 − 2 = 2. T cannot be one-to-one because nullity > 0.
This material connects directly to Section 2.2, where the action of T on a basis is encoded as a matrix. The rank of the transformation becomes the rank of the matrix. It also connects to Chapter 3 (systems of linear equations), where the null space of the associated linear transformation is the solution set of the homogeneous system. Later, in Chapters 5 and 6, eigenvalues describe directions where T acts as simple scaling, building on the foundations here.
linear map, linear function, vector space homomorphism, kernel, null space, image, range, rank, nullity, dimension theorem, rank-nullity theorem, injective, surjective, bijective, one-to-one linear map, onto linear map, rotation, reflection, projection, identity transformation, zero transformation, Friedberg Chapter 2, abstract linear algebra