In linear algebra, it’s important to know and understand how to convert a vector to a different basis because having this knowledge has various practical applications. In this article, we’ll be learning and discussing just that—from learning how to find a component vector to understanding how the change of basis matrix works. If you’re studying linear algebra and are looking to understand change of basis even more and expanding your knowledge, then you should definitely keep on reading!
Defining the Basis
Of course, in order to truly understand what the change of basis means, you must first understand what a basis is. Let’s take the vector space V—the basis of this would be a set of vectors in V that is not only linearly independent but spans V as well. An ordered basis, on the other hand, is more of a list rather than a set. In other words, what the order of the vectors is in an ordered basis is significant.
Now, let’s talk more about vector components. If you take 
as an ordered basis for 
and then 
is a vector in , you’ll find scalars 
. Thus, you have ![\[v = c_1u_1+c_2u_2+...+c_nu_n\]](https://eli.thegreenplace.net/images/math/400f6b84c3ee13d328880d7b29bb7c467c868a33.png)
. These are known as components.
How to Find a Component Vector
Now, let’s learn how to find a component vector. For this example, let’s use 
. Let’s also use two independent vectors, say, U = (2, 3), (4, 5). U would be considered as an ordered basis for . So what if we have v = (2, 4)? In this case, what would be the ![[v]_{\text{\tiny U}}](https://eli.thegreenplace.net/images/math/dc587d1ab07e4744144f02d47abbf148b6c339d4.png)
? Here’s how you can solve this problem:
![\[\begin{pmatrix} 2 \\ 4 \end{pmatrix}=c_1\begin{pmatrix} 2 \\ 3\end{pmatrix}+c_2\begin{pmatrix} 4 \\ 5 \end{pmatrix}\]](https://eli.thegreenplace.net/images/math/1793547a360ef6b1c94da80aebb1698bcd69e20e.png)
If you’re looking at this formula from a 2D perspective, this is as easy as pie to solve. The solution would become c1 = 3 while c2 = -1. Thus, would be:
![\[[v]_{\text {\tiny U}}=\begin{pmatrix} 3 \\ -1 \end{pmatrix}\]](https://eli.thegreenplace.net/images/math/548b7f57bfc01932593b5cdaa597b77f531bd03b.png)
However, the formula becomes different once we start looking for 
because it’s just like trying to solve a linear system of n equations using n variables. Because, as mentioned, the vectors from earlier are linearly independent, we can solve for by inverting a matrix.
Understanding the Change of Basis Matrix
We’ve finally reached the most important part of this article. Imagine that there are two ordered bases for one vector space, namely U = u1, u2, …, un as well as W = w1, w2, …, wn. In some cases, we’re able to find ![[v]_{\text{\tiny U}}](https://eli.thegreenplace.net/images/math/9c044f231102324a9c84edf98b7f5f37bcdc2e2e.png)
and ![[v]_{\text{\tiny W}}](https://eli.thegreenplace.net/images/math/f8ae14559a972991d3c72d1014db829284f86f6a.png)
in 
. However, how are these two related to each other? Using the examples provided above, we should be able to find the coefficients of in basis W the same way. Like before, we’ll have to convert the vectors in order for us to solve the problem correctly. Fortunately, there’s a much simpler way for you to solve this, and the key is to look for how the basis vectors of U would look in basis W. In this case, you’ll have to solve for ![[u_1]_{\text{\tiny W}}](https://eli.thegreenplace.net/images/math/f53ee2bf40857eff2038d6543b07f0cbcf02a651.png)
, ![[u_2]_{\text{\tiny W}}](https://eli.thegreenplace.net/images/math/f8f94de8a4ccd3f2d7594afd45caae1403310968.png)
and so on until ![[u_n]_{\text{\tiny W}}](https://eli.thegreenplace.net/images/math/3f2515b5f7e26704e96f99b9947392836689cf34.png)
.
Conclusion
For many people, algebra (or mathematics in general) can be a nightmare, so much so that they don’t want anything to do with numbers and solving equations. However, some people enjoy solving mathematical problems, and these people were able to create formulas and solutions to help make things easier. We hope that our article gave you more insight into the concept of the change of basis.
