the significance of matrix multiplication?

Question

the concept is just lacking of intuition for me. here is my question in the words of pivoxa15, writing in another blog:


"Matrix multiplication is clearly defined but is there a tangible or physical interpretation for it? Or is it just abstract formalism?

I am thinking about each column of the matrix as vectors so matrix multiplication with two 2by2 matrices is about multiplying 4 vectors in a certain way. The result is a 2by2 matrix with all four vectors tangled up with each other. But does the two resultant vectors in this matrix have a meaning with respect to the original vectors if we gave the original vectors some meaning in the first place?"


please don't use words like "linear map" and "standard basis" without defining them. i am fluent in multivariable calculus but i know very little linear algebra terminology.

thanks!!!

Answer 1

If you take a vector, you can transform it into some other vector using a matrix. Suppose your vector is 2 dimensional and you want to transform it into another vector in the same plane. You accomplish this by multiplying a 2x1 column vector v by some 2x2 matrix. An example of this is a rotation matrix R which will rotate your vector about the origin: v' = Rv



The components of your starting and ending vectors are determined by how you measure the coordinates in the plane. You pick a particular point as your origin and scale/orient your axes in a particular direction. However, neither the origin nor the scale/orientation of your axes are special. I can pick my own origin and scale/orient my axes however I want and whatever measurements we make should agree even thought the components of our vectors do not.



Suppose we agree on the orgin but disagree with the orientation and scale of the axes. Then there is some 2x2 matrix T that when multiplied by your vector under your coordinate system will give me that same vector in my coordinate system: w = Tv. Then in that sense, if I take your vector, multiply it by your rotation matrix and then multiply by that resulting matrix by T then I will get your rotated vector but with the components expressed in my coordinate system: w' = TRw

Answer 2

different than for the very solid familiar solutions given already, that is extra that the stress Tensor supplies you a necessary use of matrices and matrix multiplication. the stress tensor is a manner of mathematically expressing the state of stress at a element in a solid medium. Matrices are additionally used to particular the conservation equations, mass, power and momentum at each and each element of a fluid, as an occasion, in modeling how the fluid will pass interior its boundary under utilized forces and pressures. Manipulating the matrices then yields the answer ... the stress in a solid medium, or the speed distribution and different residences in a fluid medium. It gets exciting.

Answer 3

its used for translation, transformation, etc