A set of vectors is dependent when one can be written from the others, and independent when none can. The formal test: the only combination giving the zero vector uses every coefficient zero. For two vectors, dependent means one is a multiple of the other, which the determinant ad − bc detects. More vectors than dimensions are always dependent.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two vectors
⟨2, 4⟩ and ⟨3, 6⟩: ratios 3/2 and 6/4 are both 1.5, so ⟨3, 6⟩ = 1.5⟨2, 4⟩. Dependent. ⟨2, 4⟩ and ⟨3, 5⟩: ratios 1.5 and 1.25 differ. Independent.
The formal test
a⟨2, 4⟩ + b⟨3, 6⟩ = ⟨0, 0⟩ with a = 3, b = −2: ⟨6 − 6, 12 − 12⟩ = ⟨0, 0⟩. Non-zero coefficients reached zero, so the set is dependent. For an independent set, only a = b = 0 works.
The determinant
For columns ⟨a, c⟩ and ⟨b, d⟩, ad − bc = 0 exactly when one is a multiple of the other. ⟨2, 4⟩, ⟨3, 6⟩: 12 − 12 = 0, dependent. ⟨2, 4⟩, ⟨3, 5⟩: 10 − 12 = −2, independent.
Count first
Three vectors in the plane are always dependent, four in space, and so on. You cannot have more independent directions than dimensions, so counting settles the question before any arithmetic.
The zero vector spoils it
Any set containing ⟨0, 0⟩ is dependent: 1 × ⟨0, 0⟩ + 0 × everything else = ⟨0, 0⟩ with a non-zero coefficient. The zero vector adds no direction and fails the test on its own.
Why it matters
Independent vectors each earn their place: removing one shrinks the span. Dependent sets carry passengers. Chapter 6 builds bases from exactly the vectors that earn their place.
Step 2: Try It Yourself
Tap and try it out.
Test ⟨2, 4⟩ and ⟨3, 6⟩ for independence, then ⟨2, 4⟩ and ⟨3, 5⟩, by ratios, by a combination, and by the determinant
- 13/2 = 1.5 and 6/4 = 1.5: ⟨3, 6⟩ = 1.5⟨2, 4⟩a multiple: dependent
Step 0 of 4
- i-hat lands on(3, 1)
- j-hat lands on(1, 2)
- Determinant5
The shaded parallelogram is the image of the unit square, and its area is 5. That is exactly what the determinant measures.
Step 3: In Real Life
Redundant sensors
Three sensors, one of which always reads the average of the other two, give only two pieces of information. Independence says which measurements are actually earning their place.
Step 4: Watch an Example
One step at a time.
Watch Yuki Spot a Redundancy
Yuki tests ⟨1, 0⟩, ⟨0, 1⟩ and ⟨3, 5⟩ for independence.
- Step 1
She counts three vectors in a two-dimensional plane, which can hold at most two independent directions.
Step 5: Your Turn
Practice makes it stick.
The Count
Problem 1 of 2
Four vectors in the plane. Are they independent?
The Multiple
Problem 2 of 2
⟨2, 5⟩ and ⟨6, 15⟩. Are they independent?
Independent or Not
1 of 8
⟨1, 0⟩ and ⟨0, 1⟩. Independent?
2 of 8
⟨3, 6⟩ and ⟨1, 2⟩. Independent?
3 of 8
A set containing the zero vector. Independent?
4 of 8
At most how many independent vectors fit in three-dimensional space?
5 of 8
Two independent vectors in the plane have a determinant of what, if anything but zero?
6 of 8
Five vectors in three-dimensional space. Independent?
7 of 8
Match each situation to its verdict.
Tap a card on the left to start.
8 of 8
⟨4, 1⟩ and ⟨1, 4⟩. Independent?
Step 6: Quick Check
Show what you know.
Question 1 of 2
⟨2, 3⟩ and ⟨4, 6⟩. Independent?
Question 2 of 2
What does linear dependence mean?
What You Learned
- Vectors are dependent when one can be built from the others, and independent when none can.
- In the plane, two vectors are dependent exactly when one is a multiple of the other, and ad − bc = 0 detects it.
- The formal test asks whether a non-zero combination can reach the zero vector.
- More vectors than dimensions are always dependent, so count first.