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So apparently matrices work like intended...
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So apparently matrices work like intended on /sci/ now. [eqn]\begin{bmatrix}1 & 2 & 3 \\ 1 & -4 & 6 \\ 3 & -9 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0\\ 2 & -8 & 0 \\ 3 & -15 & -12 \end{bmatrix} \begin{bmatrix}1 & 2 & 3\\0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}[/eqn] Discuss
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It has been that way for at least a coupe months.
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entry (2, 1) in the matrix on the left should be = 2, not 1
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>>7784080
Sumimasen, senpai. I have failed. Sudoku is the only option at this point.
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as someone who's only used matrices so far in precalc/calc, what application does it have in physics? All I really know are how to multiply or add them together, inverses, etc. I dont see what they represent in the real world mathematically
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>>7784081
[math]
\begin{bmatrix} \cos ( \theta + \phi ) & - \sin ( \theta + \phi ) \\ \sin ( \theta + \phi ) & \cos ( \theta + \phi ) \end{bmatrix} = \begin{bmatrix} \cos ( \theta ) & - \sin ( \theta ) \\ \sin ( \theta ) & \cos ( \theta ) \end{bmatrix} \begin{bmatrix} \cos ( \phi ) & - \sin ( \phi ) \\ \sin ( \phi ) & \cos ( \phi ) \end{bmatrix}

[/math]
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>>7784086
Matrices have some application in Physics. Matrices can represent vectors. For example, the cross product of two vectors is defined as [eqn]{\bf u} \times {\bf v} = \begin{vmatrix}{\bf i} & {\bf j} & {\bf k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3\end{vmatrix}[/eqn]
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>>7784086
>what application does it have in physics?
In every single topic.
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>>7784086
You've probably so far only dealt with functions that take a real number as an input and give you a real number as an output, but there are other types. If you have a vector-valued function of a vector variable that happens to be linear (that is, f(ax+by)=af(x)+bf(y) for vectors x,y and scalars a,b) you can represent the function by a suitable matrix product f(x)=Ax. Further, all matricies represent some sort of linear function from one vector space to another (depending on the basis representation). They kind of stiff you in high school by only ever saying they're "tables of information" desu

An easy example is the "rotate counterclockwise by t radians" function from the x-y plane to itself, which is represented by the rotation matrix that does exactly that.

The physical application everyone is likely to mention is basic QM (which is actually functional analysis, but you can just think of that as "infinite-dimensional matricies" for now if youre highschool or lower-div. Szabo's Modern Quantum Chemistry has an excellent self-contained math prelims chapter that will get you up to speed on the essentials of linear algebra theory (the rest of the book is good too, though perhaps a little advanced.)
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>>7784086

You can do some funky shit with them, in fact you can group scalars, vectors, and matrices into a single category called tensors.

This is my naive engineering understanding of it
rank 0 tensor - scalar
rank 1 tensor - vector (column of numbers)
rank 2 tensor - matrix (n x n matrix of numbers)
rank 3 or higher - tensor
for rank 3, I think of it as a n x n x n cube of numbers

It was neat as fuck manipulating these things when I had to, why some operations are defined the way they are.

Can any math fag elaborate?
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>>7784106
This definition always bothered me until I realized the i, j, and k are the i, j, and k of the quaternion group.
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>>7784060
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>>7784716
By analogy with matrices as representations of linear maps, rank 3 tensors can be thought of as representations of multilinear maps. But there are different specific formulations depending on the field.
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