A 4×6 matrix A has rank 3. What is the dimension of its null space?
A3
B1
C4
DCannot be determined without more information
By the rank-nullity theorem, rank(A) + nullity(A) = n, where n is the number of *columns*. Here n = 6 and rank = 3, so nullity = 6 − 3 = 3. The common error is using the number of rows (4) instead of columns (6): students subtract rank from 4 and get 1. The number of rows tells you the dimension of the output space (R⁴), not the dimension of the input space; nullity concerns free variables in the *input* space R⁶.
Question 2 Multiple Choice
A student defines the null space as: 'the set of all x satisfying Ax = b for some right-hand side b.' What is wrong with this definition?
ANothing — this is equivalent to the correct definition
BThe null space is specifically the solutions to Ax = 0 (the zero vector), not Ax = b for a general b
CThe null space is a matrix, not a set of vectors
DThe null space only exists when A is square and invertible
The null space is exclusively the solution set of the *homogeneous* system Ax = 0. Allowing an arbitrary b on the right-hand side gives a different object: the solution set of a non-homogeneous system, which is either empty or an affine subspace (not passing through the origin). Only the homogeneous solution set is guaranteed to be a subspace — it always contains the zero vector and is closed under addition and scalar multiplication. The null space is defined by b = 0 specifically.
Question 3 True / False
If the null space of matrix A contains only the zero vector, then the linear transformation T(x) = Ax is injective (one-to-one).
TTrue
FFalse
Answer: True
Injectivity means T(x₁) = T(x₂) implies x₁ = x₂. If Ax₁ = Ax₂, then A(x₁ − x₂) = 0, so x₁ − x₂ is in the null space. If the null space is trivial (only the zero vector), then x₁ − x₂ = 0, i.e., x₁ = x₂. Conversely, a non-trivial null space directly witnesses non-injectivity: any non-zero vector in nul(A) maps to 0 along with the zero vector itself.
Question 4 True / False
The nullity of a matrix A equals the number of rows in its reduced row echelon form.
TTrue
FFalse
Answer: False
Nullity equals the number of *free variables* in the RREF of A, not the number of rows. Free variables correspond to columns without pivot positions. If A is m×n with rank r, then there are n − r free variables, so nullity = n − r. The number of rows is m and determines the dimension of the output space; it has no direct relationship to the null space dimension.
Question 5 Short Answer
A matrix A, when row-reduced to RREF, yields 4 free variables. What does this tell you about the null space, and what does it mean geometrically?
Think about your answer, then reveal below.
Model answer: The null space has dimension 4 — it is a 4-dimensional subspace of the input space. Each free variable parameterizes one independent direction in the solution set of Ax = 0. Geometrically, the null space is a 4-dimensional flat (hyperplane through the origin) consisting of all vectors that A collapses to zero.
Each free variable contributes exactly one basis vector to nul(A). You find these basis vectors by setting each free variable to 1 (and others to 0) and solving for the pivot variables. The geometric picture is important: a 4-dimensional null space means A 'destroys' 4 independent directions — it collapses a 4-dimensional family of vectors to zero. The larger the null space, the more information A discards.