MAT 242 Chap.8 Row Space, Column Space and Null Space
Row Space, Column Space and Null Space
Three subspaces attached to every matrix
Each matrix A comes with three subspaces. The null space collects the vectors that A sends to zero; the row space is spanned by the rows; the column space is spanned by the columns and consists of exactly the right-hand sides b for which Ax = b can be solved.
One reduced row echelon form gives bases for all three, but each basis is read off by a different rule, and mixing up the rules is the classic error.
Reading the bases
For the null space, solve Ax = 0, give each free variable a parameter, and split the solution into one vector per parameter; those vectors form the basis, and their number is the nullity.
For the row space, take the nonzero rows of the RREF, because row operations never change the row space. For the column space, use the RREF only to locate the pivot columns and then copy those columns from the original A, because row operations do change the column space.
Rank
The row space and column space always share a dimension, the rank, which equals the number of pivots.
Since every column is either a pivot column or a free-variable column, rank plus nullity equals the number of columns. A square matrix is invertible exactly when its rank is full.
The paid chapter computes all five quantities for one matrix, then shows that the solutions of Ax = b are one particular solution plus the null space.
The course's written homework asks for exactly this bundle of five answers on one matrix, so the chapter ends with a summary grid of the three basis rules.
How sample solutions grade this
Sample Test 2 solutions award separate credit for each of the five answers and deduct for the familiar mix-ups: forgetting a variable in the null space, taking column-space vectors from the RREF, taking row-space vectors from the original matrix, and using non-pivot columns.
A column of zeros in A still represents a variable, and it is free, so it must appear in every null-space vector. Checking rank plus nullity against the number of columns catches a missing variable immediately.
Particular plus homogeneous
The solutions of a system with a nonzero right-hand side are not a subspace, but they have a clean structure: one particular solution plus every vector of the null space.
The course notes point out that this is the same split used for differential equations, where the null-space part is called the homogeneous solution. Recognising the split makes it easy to read the dimension of a solution family straight from the nullity.
What this chapter covers
- 01
Null space and nullity
- 02
Row space from the reduced rows
- 03
Column space from the original pivot columns
- 04
Rank and full rank
- 05
Rank plus nullity equals the number of columns
Worked example · free
Rank, nullity and a null-space basis
- 2Pivots sit in columns 1 and 3, so rank A = 2 and x₂, x₄ are free.
- 2With x₂ = s, x₄ = t: x₁ = −3s + 4t and x₃ = −6t.
- 2Split by parameter: basis {(−3, 1, 0, 0), (4, 0, −6, 1)}, so nullity 2 and 2 + 2 = 4 columns.
Key terms
- Null Space
- The null space of A is the set of all vectors x with Ax = 0.
- Column Space
- The column space of A collects every combination of its columns, equivalently every b for which Ax = b has a solution.
- Rank
- Rank is the common dimension of the row space and column space, equal to the number of pivots.
- Nullity
- Nullity is the dimension of the null space, equal to the number of free variables.
Row Space, Column Space and Null Space FAQ
Why take column-space vectors from the original matrix instead of the RREF?
Row operations change the column space, so the RREF's columns usually span a different subspace. The RREF is only used to locate the pivot positions; the basis vectors come from A itself.
What does rank plus nullity tell me?
It always equals the number of columns, because each column is either a pivot column or a free-variable column. It is a fast consistency check on every null-space and rank answer.
Is the row space the same as the column space?
They have the same dimension, the rank, but they usually live in different spaces: rows of an m × n matrix are vectors with n entries, columns have m entries.
Exam move
Memorise the three basis rules as a table, then practise on wide matrices with more columns than rows; those are the ones where free variables and dimension checks matter most. After each practice matrix, write the five answers in a fixed order: null space, row space, column space, rank, nullity. The fixed order prevents forgetting one.
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