Dimension and the Rank-Nullity Theorem
Every basis of a space has the same size, which is its dimension. For a matrix, the rank and the nullity add to the number of columns, which explains the shape of every solution set met so far.
What a learner can do afterwards
- Find a basis for the column space and for the null space of a matrix
- State the rank-nullity theorem and check it on an example
- Predict the number of free parameters in a solution set from the rank
1 · Read
You already know what a basis is. The dimension of a space is the number of vectors in any basis. Every basis of one space has the same size, so one number tells how big the space is and how many coordinates each vector needs.
For a matrix, rank is the dimension of the column space, counted by pivot columns. Nullity is the dimension of the null space, counted by free variables, one vector each. An independent spanning pair, such as (1, 1, 0) and (0, 1, 1), is a basis of size 2.
Take a 3 by 5 matrix with pivots in columns 1 and 3. Rank is 2, so a column space basis has 2 vectors. Free variables x2, x4 and x5 give nullity 3. Rank plus nullity is 2 plus 3, which is 5, the column count.
A 5 by 5 matrix of rank 5 has nullity 0, so Ax equals 0 has only the zero solution and Ax equals b has exactly one solution for every b. Contrast [[1, 3], [2, 6]]: the second column is 3 times the first, so both columns lie on one line and the column space is a line, not all of R squared. Rank can never pass the row count or the column count.
Forecast free parameters with columns minus rank, since nullity counts them. Six unknowns with rank 4 means 2 free parameters. A 3 by 6 matrix can never reach rank 4, since 3 rows cap it at 3.
Rank plus nullity equals the column count, and the nullity tells you the number of free parameters.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.