2. Representing Vectors

Linear Algebra



Weiqi Zhou

Motivation

  • let be linearly dependent

  • i.e.,

  • w.l.o.g., say

  • then

Linear Dependence Representable by Linear Combinations

for the component with non-zero coefficient

Maximally linearly independent set

  • let be linearly independent, if are linearly dependent for any , then is said to be a maximally linearly independent set on

  • examples：,

  • non-examples：,

Dimension and bases

  • if is a maximally linearly independent set on ，then

  • if is a maximally linearly independent set on ，then is said to be a set of basis on

Any vector in can be written as a unique linear combination of basis vectors

Existence

  • take an arbitrary ，then are linearly dependent

  • there exist so that , in particular

  • therefore

Uniqueness

  • suppose that
    are two different linear combinations, then
    。

  • consequently one gets

  • contradiction

Rank

  • given a set of vectors , and that are linearly independent

  • if are linearly dependent for any , then is the rank of , i.e., the rank of (if finite) is the size of the maximally linearly independent set in

  • if , then is said to be full rank

Rank/dimension is independent of the choice of the maximally linearly independent set

Exercise

  • determine ranks of the following two sets of vectors respectively



Span

  • the set of all linear combinations of vectors in is called the span of

  • If is a maximally linearly independent set in

  • then every vector in can be written as a unique linear combination of

  • the span of coincides with the span of

Connections to linear systems

  • There are some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?



  • alternatively:

Solving a linear system is to find whether (and how) the vector in the RHS can be written as a linear combination of vectors in the LHS

The unique solution case

  • given , if is in the span of , and is linearly independent, then there is a unique solution

  • example：

The infinitely many solution case

  • given , if is in the span of , but is linearly dependent, then there are infinitely many solutions

  • example：

The no solution case

  • given , if is not in the span of , then there is no solution

  • example：

Summary

  • motivation

  • maximally linearly independent sets

  • structure of solutions to linear systems