Practice question · Put in order
Order the steps of computing a basis for the kernel of a matrix A.
- Read off one basis vector of the kernel for each free variable
- Write down the homogeneous system Ax = 0
- Row reduce A to echelon form
- Express every solution in terms of the free variables
- Identify the pivot columns and hence the free variables
Hints
- The kernel is by definition a solution set, so start by writing that system.
- The number of free variables turns out to be the nullity.
Show the answer
- Write down the homogeneous system Ax = 0
- Row reduce A to echelon form
- Identify the pivot columns and hence the free variables
- Express every solution in terms of the free variables
- Read off one basis vector of the kernel for each free variable
Why
Each free variable contributes one basis vector of the kernel, so nullity equals the number of free variables and rank equals the number of pivots. Since every column is either a pivot column or a free column, the two counts add to the number of columns, which is Rank-Nullity, proved by bookkeeping.
Practise Kernel, Image, and Rank–Nullity
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