Basis

Let be a vector space over a field , and


Definition (Linearly dependent vectors)

are defined to be linearly dependent if not all zeros, such that

 


Definition (Linearly independent vectors)

are defined to be linearly independent if

 
Remark

Let .

is a set of linearly independent vectors if

 
Definition (Basis)

A set is defined as basis if:

  • is a set of linearly independent vectors
 


Theorem (Existence and uniqueness of the coordinates)

Let be an ordered basis for over . Then, and uniquely determined some scalars such that , . The n-tuple is the coordinate vector of relative to the basis .

 


Proof

  • Existence: such that
  • Uniquely determined: let's suppose (Reductio ad absurdum) that the following expressions are both true, for some , with , such that :

Subtracting the second to the first one, the result is:

All the are linearly independent (because they are vectors of a basis), therefore it follows that , which contradicts the hypothesis. Thus, the are uniquely determined. }}

 


Theorem (Linearly dependent set of vectors)

is linearly dependent such that is linear combination of the remaining .

 


Proof

is linearly dependent such that , where are not all s. Without loss of generality: Therefore is linearly dependent. }}

 


Definition (Maximal set of linearly independent vectors)

Let . Then is a maximal subset of linearly independent vectors if:

  • linearly independent
  • is linearly dependent
 


Theorem (Basis as maximal set of l.i. spanning set of vectors)

Let be a spanning set for , and be a maximal set of linearly independent vectors (where ), then is a basis.

 
Proof

I need to prove that spans , which means I have to prove that such that .

By hypothesis, such that . , is linearly dependent. Therefore, every can be expressed as linear combination of the vectors in .

It follows that every can be expressed as linear combination of the vectors in , so, is a basis.

 


Theorem

Let be a vector space over and let be a basis for . If and , then are linearly dependent.

 
Proof

If are linearly dependent, the theorem is trivially proved, so let's proceed in the proof assuming they are linearly independent.

Let's prove by induction that .

The induction hypothesis is: where such that (after having possibly rearranged ), the vectors will span .

Base case:

is a basis, therefore such that:

and WLOG , so:

spans , and , therefore the last set spans as well.

Inductive step:

For the induction hypothesis, . Therefore, such that

Not all can be zero because we assumed all to be linearly independent so, WLOG, and we get to the result:


It is now proved by induction that It follows that such that That is, are linearly dependent.

 
Remark

From this theorem, it trivially follows that all the possible bases for a particular vector space, have the same cardinality. In fact, suppose that a vector space has two possible bases, the first one has elements, the second one has elements. It can be neither nor , therefore

 
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