Subject GRE-Style: Rank and Nullity

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Rank and Nullity Finding the rank and null space of a matrix Question: What is the rank of , and what is the dimension of its null space? Observation: There is exactly one nonzero row (the first), so the rank is 1. Apply the Rank-Nullity Theorem We have a matrix with rank 1. The rank-nullity theorem gives: The null space is 2-dimensional. Understanding the null space: Notice that rows 2 and 3 are identical to row 1. The matrix is rank 1 because only one row (and one column) is "independent." The null space consists of all vectors satisfying: This is a 2-dimensional subspace (since we have one constraint on 3 variables). Key lesson: Use rank-nullity to find nullity quickly. You don't always need to solve the system explicitly!

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