basis

assuming a vector space , let be a finite, linearly independant and spanning set of which we call a basis, each vector in a basis is called a basis vector

let be a basis of :

since is linearly independant then: this is necessarily true because has to be 0 for to zero out and if it wasnt zero then the output vector would have a non-zero in it and it wouldnt be anymore

a basis doesnt have to be of this simplified form, we can take the basis , apply elementary row operations to its vectors and we would get another basis for the given vector space

every finitely generated vector space has atleast one basis

since is finitely generated there exists a finite set such that

according to [BROKEN LINK: can_drop_vector_keep_span], there exists where is linearly independant and , therefore is a basis

is linearly independant is a basis of

given the vector space , we find a basis of it: so is a basis of and , see dimension

consider the vector space which represents all the matrices with dimensions over the field

what we are basically looking for is a set of matrices that is linearly independant so that no matrix is a linear combination of the others, which is accomplished with: and: we prove that is linearly independant: it must be that and so is linearly independant because the only linear combination that gives us is where the coefficients all are 0

given is a finite set and and , then is linearly dependant

in other words, if we have a finitely generated vector space, if is a subset of that space such that the number of elements in is bigger than the number of elements in , then is linearly dependant

let and such that

let denote the coefficients, for every where there exist the coefficients:

let be a finite set, if and is linearly independant then

let be bases of then

is a spanning set and and is linearly independant then is a spanning set and and is linearly independant then therefore

basis of the cartesian coordinate system

the of the cartesian coordinate system consists of the unit vectors that lie along the x, y, and z axes

the x unit vector is denoted by , the y unit vector by , and the z unit vector by

we can write any vector in terms of the base vectors