Linear dependence and the null space

Drag either 3D picture to rotate it.

1. If C lies in span(A, B), does B lie in span(A, C)?

Blue sheet: span(A, B). Green sheet: span(A, C), shown when ticked.
Build C as
C = a·A + b·B + h·(up, off the sheet)
a
b
h
The rule Dependence belongs to the whole set, not to one vector. The set {A, B, C} is dependent when some combination with not-all-zero weights gives 0:
x·A + y·B + z·C = 0
Any vector whose weight is non-zero can be moved to the other side and rebuilt from the rest. A vector whose weight is 0 cannot.

Pairs are not enough With C = A + B, no two of the three vectors are parallel, yet all three are dependent. Test all of them together: for n vectors in ℝⁿ, dependent means det = 0; otherwise row-reduce.

2. The null space

vectors on the null line   a parallel line   where î lands   where ĵ lands
Matrix (columns are where î and ĵ land)
morph t
Name The vectors that land on 0 form the null space of the matrix, also called the kernel of the transformation. 3Blue1Brown covers it in chapter 7.

How to find it Solve A·x = 0 by row reduction. Each free variable gives one direction of the null space.

Why it matters Every solution of A·x = b is one particular solution plus anything from the null space. A non-zero null space, which is the same thing as a free variable, means infinitely many solutions.

The link to part 1 A null vector is a dependence among the columns. In the default matrix, (−2, 1) is in the null space because −2·col₁ + 1·col₂ = 0. That is also the rank count: rank + (dimension of null space) = number of columns.