Linear Independence

Which vectors are actually earning their place.

12 problems · about 23 minutes · N-VM.B.4, N-VM.B.5

What this lesson teaches

The student tests a set of vectors for linear independence and interprets the result.

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.

Straightforward practice. Get the method working first.

5 problems

⟨2, 3⟩ and ⟨4, 6⟩. Independent? 1 yes, 0 no.
Answer 0
Why No.
What does linear dependence mean?
Answer At least one vector can be built from the others, so it adds nothing.
Why One is expressible from the rest.
⟨1, 0⟩ and ⟨0, 1⟩. Independent? 1 yes, 0 no.
Answer 1
Why Neither is a multiple of the other.
⟨3, 6⟩ and ⟨1, 2⟩. Independent? 1 yes, 0 no.
Answer 0
Why Triple the second.
A set containing the zero vector. Independent? 1 yes, 0 no.
Answer 0
Why The zero vector always spoils it.

The same ideas with more to keep track of.

3 problems

At most how many independent vectors fit in three-dimensional space?
Answer 3
Why One per dimension.
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.
Answer 1
Why Zero determinant means dependence.
Five vectors in three-dimensional space. Independent? 1 yes, 0 no.
Answer 0
Why More vectors than dimensions.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each situation to its verdict.
Answer One vector is a multiple of another → Dependent; More vectors than dimensions → Dependent; The determinant is non-zero → Independent
Why Only one of the three signals independence.
⟨4, 1⟩ and ⟨1, 4⟩. Independent? 1 yes, 0 no.
Answer 1
Why Neither is a multiple of the other.
The Count: Four vectors in the plane. Are they independent? 1 yes, 0 no.
Answer 0
Why No.
The Multiple: ⟨2, 5⟩ and ⟨6, 15⟩. Are they independent? 1 yes, 0 no.
Answer 0
Why No.