A set of vectors is linearly dependent when one of them can be written using the others. That one adds nothing to the span.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Independent
A set is linearly independent when no vector is a combination of the rest. Every one contributes a genuinely new direction.
The formal test
Independence means the only way to combine them into the zero vector is to take every coefficient to be zero.
The two-vector case
Two vectors are dependent exactly when one is a multiple of the other. That is the whole test in the plane.
A counting limit
Three vectors in the plane are always dependent. You cannot have more independent directions than dimensions.
The zero vector spoils it
Any set containing the zero vector is dependent, because that vector can be scaled by anything and change nothing.
Which vectors earn their place
A set is linearly dependent if one vector is a combination of the others, so it adds no new reach. Independence means every vector contributes a direction the others cannot produce.
The formal test
Vectors are independent if the only combination giving the zero vector uses all-zero coefficients. That is checked by solving a homogeneous system, so independence reduces to elimination.
Too many vectors are always dependent
More than n vectors in n-dimensional space must be dependent, because there is not enough room for that many independent directions. Counting alone settles many cases immediately.
Why it matters
A dependent set contains redundancy, so a description built from it is not minimal. Independence is what makes a basis the smallest possible description, which is the point of the concept.
Step 2: Try It Yourself
Tap and try it out.
- 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: 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 living in a two-dimensional plane.
Step 4: Your Turn
Practice makes it stick.
The Count
Problem 1 of 2
Four vectors in the plane. Are they independent? 1 yes, 0 no.
The Multiple
Problem 2 of 2
⟨2, 5⟩ and ⟨6, 15⟩. Are they independent? 1 yes, 0 no.
Independent or Not
1 of 8
⟨1, 0⟩ and ⟨0, 1⟩. Independent? 1 yes, 0 no.
2 of 8
⟨3, 6⟩ and ⟨1, 2⟩. Independent? 1 yes, 0 no.
3 of 8
A set containing the zero vector. Independent? 1 yes, 0 no.
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? Enter 0 if it must be zero, 1 if it must not be.
6 of 8
Five vectors in three-dimensional space. Independent? 1 yes, 0 no.
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? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
⟨2, 3⟩ and ⟨4, 6⟩. Independent? 1 yes, 0 no.
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.
- You can never have more independent vectors than there are dimensions.