Linear Independence

Which vectors are actually earning their place.

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

NameDate

Figure — use these to answer the problems

A determinant of zero is exactly linear dependence. Line the two arrows up and watch it drop to zero.

first vector, entry 13

second vector, entry 11

first vector, entry 21

second vector, entry 22

Figure 1

Straightforward practice. Get the method working first.

5 problems

⟨2, 3⟩ and ⟨4, 6⟩. Independent? 1 yes, 0 no.
Answer
2
What does linear dependence mean?
- At least one vector can be built from the others, so it adds nothing.
- All the vectors are equal.
⟨1, 0⟩ and ⟨0, 1⟩. Independent? 1 yes, 0 no.
Answer
⟨3, 6⟩ and ⟨1, 2⟩. Independent? 1 yes, 0 no.
Answer
A set containing the zero vector. Independent? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

At most how many independent vectors fit in three-dimensional space?
Answer
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
Five vectors in three-dimensional space. Independent? 1 yes, 0 no.
Answer

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

4 problems