How this instrument works
A matrix's null space is the set of every vector that the matrix sends to zero — everything the transformation completely erases. Its dimension, called the nullity, measures how large that erased set is. For a 2×2 matrix, the Rank-Nullity Theorem gives a direct shortcut: nullity = 2 − rank, since the total of a matrix's rank (what survives) and its nullity (what's erased) must always equal the number of columns, 2 here.
A full-span 2×2 matrix (rank 2) has a nullity of exactly 0 — only the zero vector itself maps to zero, meaning the transformation loses no information at all. A matrix scoring 1 on that same scale has a nullity of 1 instead — an entire line's worth of vectors gets squashed down to zero. And the zero matrix, with a span of 0, has the maximum possible nullity, 2 — every single vector in the plane maps to zero.
This is a genuinely different question from this site's Matrix Rank page, even though both start from the identical underlying calculation: that page asks how much SURVIVES the transformation, while nullity asks how much gets ERASED — two complementary views of the same matrix, always summing to the same fixed total.
- Enter the matrix's four entries into the a, b, c, and d fields, reading left to right, top to bottom.
- Read Nullity: the sheet finds the rank first, then applies the Rank-Nullity Theorem.
- Add the nullity back to the rank to confirm the total always comes out to exactly 2.
Worked example — [[2,1],[3,4]]
The matrix [[2,1],[3,4]] has full rank, 2 (its determinant, 5, is nonzero), so its nullity is 2−2=0 — only the zero vector itself maps to zero under this matrix; nothing else is lost.
The zero matrix [[0,0],[0,0]] has rank 0, giving a nullity of 2−0=2 — every vector in the plane maps to zero, the maximum possible nullity. The matrix [[2,4],[1,2]] has rank 1 (its rows are proportional), giving a nullity of 2−1=1 — a whole line's worth of vectors gets squashed to zero, consistent with the Rank-Nullity Theorem's fixed total of 2.
Questions
What is the null space of a matrix?
The set of every vector that the matrix transforms into the zero vector — everything the transformation completely erases. Its dimension is called the nullity.
What is the Rank-Nullity Theorem?
It states that a matrix's rank plus its nullity always equals the number of columns. For a 2×2 matrix, that total is always exactly 2, so nullity = 2 − rank directly.
What does a result of 0 mean?
Only the zero vector maps to zero — the matrix loses no information at all, and it's invertible (full rank). This is the smallest possible value for this figure.
What does the maximum value of 2 mean?
Every vector in the plane maps to zero — this only happens for the zero matrix itself, which has rank 0 and therefore the largest possible score on this scale for a 2-column matrix.
How is this different from the Matrix Rank page?
Both start from the same underlying rank calculation, but rank measures how much of the plane SURVIVES the matrix's transformation, while nullity measures how much gets ERASED — complementary numbers that always sum to the same fixed total.