Interactive Linear Algebra
Dan Margalit & Joseph Rabinoff · Georgia Tech · Open textbook
Read §2.5 for independence and §2.7 for basis and dimension.
Open the sourceIndependence: when a choice adds something
A new label is not necessarily a new direction.
Prerequisites: 06 · Coordinates are not the arrow
Understand
Objectives
- Find a nontrivial zero combination for three plane vectors.
- Distinguish pairwise independence from joint independence.
- Test whether removing a vector leaves a minimal dependency.
Start with something you can see
Take arrows a=(1,0), b=(0,1), and c=(2,3). Predict whether adding c gives a third independent direction. In the plane it does not: c=2a+3b. Every pair can still be independent. The short MIT passage asks whether there is a nontrivial way to make zero at 5:40. [1]
Give the idea a precise name
The span of selected vectors is the set of all their linear combinations. A selection v₁,…,vₖ is linearly independent if its only combination giving the zero vector uses zero for every coefficient. A basis is an independent set spanning the whole vector space; its size is the dimension. A dependency is minimal when removing any member destroys that dependency.
c₁v₁+⋯+cₖvₖ=0 ⇒ c₁=⋯=cₖ=0
The v symbols name vectors, each c is a scalar, and the zero on the left is the zero vector, not a statement that each selected vector is zero. The all-zero coefficients always work; the question is whether another choice also works. Different arrows can still be dependent.
Work one small world
With a=(1,0), b=(0,1), c=(2,3), the equation 2a+3b−c=(0,0) uses nonzero coefficients. Each pair is independent: neither vector in any pair is a scalar multiple of the other. Removing any one leaves an independent pair, so {a,b,c} is a minimal dependency. The span is ℝ² and has dimension two.
| Selection | Independent? | Witness |
|---|---|---|
| {a,b} | yes | horizontal and vertical |
| {a,c} | yes | c has nonzero vertical part |
| {b,c} | yes | c has nonzero horizontal part |
| {a,b,c} | no | 2a+3b−c=0 |
Counterexample to “distinct vectors are independent”: a=(1,0) and d=(2,0) are different but d−2a=0. A set containing the zero vector is also dependent: coefficient 1 on that vector and 0 on all others already makes zero.
THE BRIDGEThe pattern “pairs allowed, triple forbidden” will reappear for three edges around a triangle.
WHERE THIS IDEA STOPSLinear independence of vectors is not statistical independence of random variables. Pairwise independence does not imply joint linear independence.
Experiment
List a,b,c in the plane and predict whether each pair or the full triple adds a direction. Solve the zero-combination equation before checking the table.
Can every pair be independent while the whole selection is not?
Check & explain
For a=(1,0), b=(0,1), c=(3,4), which is correct?
- All three are independent
- Every pair is independent, but the triple is dependent
- The pair {a,c} is dependent
What is the dimension of span{(2,1),(4,2),(0,3)}?
Teach back
Give a nonzero coefficient equation for the triple a,b,c, then show why no pair has such an equation. Transfer the test to {a,(2,0)}.
- Write 2a+3b−c=0 or an equivalent relation.
- Check each pair, not just the triple.
- Show an explicit dependence for the parallel pair.
Read deeper
Mathematics for Machine Learning
Deisenroth, Faisal & Ong · Freely readable textbook
Read §§2.5–2.6, printed pp.40 and 44, for independence and basis.
Open the source9. Independence, Basis, and Dimension
Gilbert Strang · MIT OpenCourseWare · Lecture · 50 min
Start at 4:30. Watch the short core 5:40–8:40; then the longer passage 4:30–29:30. The full 50-minute lecture is available from the same link.
Open the source