Rank-nullity theorem
Objective. Use dimension arguments for linear maps.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
The rank-nullity theorem states that for a linear map T: V → W with V finite-dimensional, dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). This links the dimension of the domain to how much the map collapses (nullity) versus how much image it spans (rank). For T: ℝ³ → ℝ² with matrix [[1,0,1],[0,1,1]], the rank is 2 since the two rows are independent, so nullity = 3 − 2 = 1. This theorem is a powerful counting tool for determining injectivity or surjectivity without fully solving systems.
Worked example
Use the rank-nullity theorem to find the nullity of the linear map given by matrix [[1,2,3],[2,4,6]] acting on ℝ³.
- 1Observe row 2 is exactly twice row 1, so the rows are linearly dependent.
- 2Thus rank of the matrix is 1, since only one independent row remains.
- 3The domain has dimension 3, so by rank-nullity, nullity = 3 − rank = 3 − 1.
- 4This gives nullity = 2, meaning the kernel is two-dimensional.
Answer: nullity = 2
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Start practisingCommon mistake. Students compute rank from the number of rows or columns of a matrix rather than the number of linearly independent rows (or columns), leading to incorrect nullity values when rows are dependent.
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