Why would I want to use a matrix(2)? Previous explanations have stated where matrices are applied. That doesn't answer the question: what's the point of them, when surely you could just use algebraic expansions of them in the first place? It seems a strange way of representing basic algebraic expressions in a bizarre way to just to make things difficult. I've obviously missed something. Why use them? Why was this method of representing expessions devised? Why represent expessions in such an apparently unintuitive fashion?
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closed as not a real question by M Turgeon, Grigory M, Austin Mohr, Simon Markett, Mark Bennet Aug 30 '12 at 22:01
It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.
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Because they're an easy way to represent linear transformations on finite dimensional vector spaces. |
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Do you really want to deal with expressions in $n^2$ variables? By example: you have a graphs with $n$ vertices. Calculations of shortest paths (think: directions from Google Maps) and other quantities often use matrix multiplication, where the matrices represent the graph. Now. Do you want to deal with something like: $G^2$ or: $$\begin{bmatrix} {g_{1,{1}}}^{2}+g_{1,{2}}g_{2,{1}}+g_{1,{3 }}g_{3,{1}}&g_{1,{1}}g_{1,{2}}+g_{1,{2}}g_{2,{2}}+g_{1,{3}}g_{3,{2}}&g _{1,{1}}g_{1,{3}}+g_{1,{2}}g_{2,{3}}+g_{1,{3}}g_{3,{3}} \\ g_{2,{1}}g_{1,{1}}+g_{2,{2}}g_{2,{1}}+g_{2,{3}}g_ {3,{1}}&g_{1,{2}}g_{2,{1}}+{g_{2,{2}}}^{2}+g_{2,{3}}g_{3,{2}}&g_{2,{1} }g_{1,{3}}+g_{2,{2}}g_{2,{3}}+g_{2,{3}}g_{3,{3}}\\ g _{3,{1}}g_{1,{1}}+g_{3,{2}}g_{2,{1}}+g_{3,{3}}g_{3,{1}}&g_{3,{1}}g_{1, {2}}+g_{3,{2}}g_{2,{2}}+g_{3,{3}}g_{3,{2}}&g_{1,{3}}g_{3,{1}}+g_{2,{3} }g_{3,{2}}+{g_{3,{3}}}^{2} \end{bmatrix} $$ This is just a $3\times 3$ matrix, raised to only power $2$. You have families of $n^2$ variables related together in an obvious pattern: some sort of linearity. Do you want to ignore these properties and patterns and just deal with messy expressions of $n^2$ variables? Or, more elegantly, resort to an abstraction which replaces this mess. Moreover, matrices (considered as numbers in a system of matrices) turned out to form interesting properties: see matrix ring; not to mention vector spaces etc. |
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Matrices are useful in a number of theoretical aspects and as a computational aid. Fixing a basis for a vector space, matrices are very convenient ways to represent linear transformations. Moreover, they are very useful for computing compositions of linear transformations (such as computing the product of matrices by the multiplying across row of the first matrix and down the column of the other matrix method). Solving systems of linear equations are often very useful in numerous applied setting. There are many useful tricks such as row reduction to find solutions. In a more theoretical setting, often you may want to determine if linear transformations are similar, i.e. given linear transformation $A$ and $B$ does there exists a $C$ such that $CAC^{-1} = B$. One possible way of doing this, rather than trying to find this $C$ used to conjugate, is to try to put these matrices representation into rational canonical form. |
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