In linear algebra, we may consider some finite-dimensional vector space, which can have associated with it some basis with which we can work with respect to. Normally, the standard basis suffices, but it may assist us to change basis in order to transform certain problems to simpler ones. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

Ideas

Say we have a finite-dimensional vector space V with dim V = n. Then suppose there are bases B₁ = {e₁, ..., e_n}, B₂ = {f₁, ...., f_n} Now, we have

f₁ = a₁₁e₁+a₁₂e₂+...+a_1ne_n

f₂ = a₂₁e₁+a₂₂e₂+...+a_2ne_n

:

f_n = a_n1e₁+a_n2e₂+...+a_nne_n

Clearly, we can create a matrix from these equations:

This is known as a change-of-basis matrix.

If we have a vector v with coordinates in B₁, we can change the vector to have coordinates in B₂ by multiplication with M, ie, M v. We can see that this is the case in seeing that [v]_B1=b₁e₁+...+b_ne_n and the change-of-basis matrix is clearly linear.