let be linearly dependent

i.e.,

w.l.o.g., say (otherwise swap and where )

then

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

example： or on

non-example： on or on

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

suppose that
are two different linear combinations, then

consequently , contradiction

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

determine ranks of the following two sets of vectors respectively

If is a maximally linearly independent set in

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

the set of all linear combination of is called the span of

the span of coincide with the span of

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

alternatively:

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

example：

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

example：

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

example：

motivation

maximally linearly independent sets

structure of the solution to a linear system