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Matrix multiplication

Overview#

Matrix multiplication or the matrix product is a binary operation that produces a matrix from two matrices.

The definition is motivated by linear equations and linear transformations on vectors, which have numerous applications in applied mathematics, physics, and engineering.

More Detail#

if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a column of B and summed to produce an entry of AB. When two linear transformations are represented by matrices, then the matrix product represents the composition of the two transformations.

A simple example consider the Linear transformation that rotates space 90 degrees around the y axis so that would imply that it takes iHat to the coordinates 0 0 negative 1 on the z axis it doesn't move J hat so it stays at the coordinates 0 1 0 and then kHat moves over to the x axis at 1 0 0 those three sets of coordinates become the columns of a matrix that describes that rotation transformation to see a wear vector with coordinates XYZ lands the reasoning is almost identical to what it was for two dimensions each of those coordinates can be thought of as instructions for how to scale each basis vector so that they add together to get your vector and the important part just like the 2d case is that this scaling and adding process works both before and after the transformation so to see where your vector lands you multiply those coordinates by the corresponding columns of the matrix and then you add together the three results multiplying two matrices is also similar whenever you see two 3x3 matrices getting multiplied together you should imagine first applying the transformation encoded by the right one then applying the transformation encoded by the left one it turns out that 3d matrix multiplication is actually pretty important for fields like computer v = xiHat + yjHat + zkHat

Matrix multiplication/vector-multiplication.png

More Information#

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