# Application of linear transformation and matrices

I have been studying linear algebra for some time now, and I have seen some really interesting applications of linear transformation and matrices such as finding the integral $$\int x^2e^x dx$$. Does anyone have some other interesting or unexpected uses of linear algebra in analysis or any other field of mathematics?

• A few examples: a lot of graph theory uses linear algebra, representation theory is a linear-algebraic approach to abstract algebra, quantum mechanics uses linear algebra heavily, the study of differential equations and partial differential equations leans heavily on linear algebra Commented Jan 8, 2020 at 15:28
• Could you tell us more about this trick for finding that integral? Commented Jan 8, 2020 at 15:29
• If you define a basis B as $B=(e^x, x e^x ,x^2 e^x)$, and you write and define $D:B \rightarrow B$, where D is a differential operator, then you can find a matrix for that transformation, and inverse of that matrix represents integration. Commented Jan 8, 2020 at 15:34
• Oh, that's nifty Commented Jan 8, 2020 at 15:39
• I know right :) Commented Jan 8, 2020 at 15:41

One impressive example is the application of linear algebra to the study of Markov chains, which is a combination of probability and graph theory. In a nutshell, a Markov chain is a graph on which we "hop" between vertices. At any given moment, the probability of hopping from vertex $$i$$ to vertex $$j$$ is given by a (constant) probability $$p_{i,j}$$.
As it turns out, the "probability-matrix" $$P$$ whose $$i,j$$ entries are $$p_{ij}$$ encapsulates the behavior of a Markov chain very well. In particular, it turns out that the $$i,j$$ entry of $$P^k$$ is the probability that, in $$k$$ steps, we transition from vertex $$i$$ to vertex $$j$$.
An important result is that for a Markov chain (satisfying certain assumptions) with probability matrix $$P$$, the stationary distribution of the Markov chain can be computed as the eigenvector of $$P^T$$ corresponding to $$\lambda = 1$$. As it so happens, it is precisely this trick which is the main idea behind the PageRank algorithm (Google's first search algorithm, which is a big part of how the company became successful).