From: 3blue1brown
The concept of null space is described through the visual lens of linear transformations [00:00:14].
Definition and Intuition
The null space of a matrix is the set of all vectors that, after a linear transformation, land on the origin (the zero vector) [00:10:32]. It can also be referred to as the “kernel” of the matrix [00:10:37]. This means it’s the space of all vectors that become “null” by landing on the zero vector [00:10:39].
Null Space in Transformations
When a matrix is not full rank (meaning it squishes space to a smaller dimension), many vectors can land on the origin [00:09:52].
- 2D to 1D: If a 2D transformation squishes space onto a line, there is a separate line of vectors that get squished onto the origin [00:10:01].
- 3D to 2D: If a 3D transformation squishes space onto a plane, there is also a full line of vectors that land on the origin [00:10:11].
- 3D to 1D: If a 3D transformation squishes all of space onto a line, then a whole plane full of vectors lands on the origin [00:10:20].
Relation to Linear Systems of Equations
In the context of a linear system of equations, when the constant vector V (on the right-hand side of Ax = V) is the zero vector, the null space provides all possible solutions to that equation [00:10:45]. Understanding the null space helps to comprehend the set of all possible solutions [00:11:17].