Linear Dependence and Independence

Question 1

Define linear dependence.

Answer 1

Let V be a vector space and u1,···,um∈V. Then the
set {u1,···,um} is Linearly Dependent (LD) (or we say
that u1,···,um are LD) if and only if there are scalars
a1,···,am∈R and not all of them are zero, such that
a1u1+a2u2+···+amum=0.

Question 2

What does it mean for a set to be linearly dependent?

Answer 2

There exists a non-trivial solution to the dependence relation.
It is the generalization of collinear (parallel) and coplanar vectors in R^n.

Question 3

Define linear independence.

Answer 3

Let V be a vector space and u1,···,um∈V. Then the set {u1,···,um} is Linearly Inependent (LI) (or we say that u1,···,um are LI) if and only if the only solution to the dependence relation a1u1+a2u2+···+amum=0 is the trivial solution a1=a2=…=am=0

Question 4

What does it mean for a set to be linearly independent?

Answer 4

The trivial way 0u1+0u2+···+0um=0 is the only way to express 0 as a linear combination of u1,···,um.
It is the generalization of non-collinear and non-coplanar vectors in R^n.

Question 5

What are 8 facts about LI and LD?

Answer 5

1. {0} is LD,
2. Any set containing 0 is LD,
3. Any set consisting of just one vector v is LI ⇔ v≠0,
4. {u,v} is LD ⇔ one of the vectors is a multiple of the other,
5. A set with three or more vectors could be LD even if
   no two vectors are multiple of one another,
   Example: {(1, 0),(0, 1),(1, 1)} is LD,
6. If {u1,···,um} is LD, then any set containing u1,···,um is
   also LD,
   Example: {(2, 0),(3, 0)} is LD, so is {(2, 0),(3, 0),(0, 1)},
7. If {u1,···,um} is LI, then any subset of {u1, · · · , um} is LI,
   Example: {(1, 0, 0),(0, 1, 0),(0, 0, 1)} is LI, so is
   {(1, 0, 0),(0, 1, 0)},
8. {u1,···,um} is LD ⇔ at least one of the m vectors is a
   linear combination of the others,