Saturday, November 19, 2011

How to conceptualize matrix-vector multiplication in Linear Algebra?

I understand how to multiply a matrix and a vector to get a new vector. But I don't understand what is happening or what it really means to do this. Kind of a vague question but I don't know how else to describe it.|||Basically, any linear transformation can be written in the form of a matrix. So matrix-vector multiplication can be thought of as taking a vector and linearly transforming it to another vector (the solution).





There are three types of elementary matrices, those that rotate, those that stretch, and those that reflect. You can break any matrix down into these elementary matrices such that their product is the original matrix. The elementary matrices then give a step-by-step guide as to how the vector is being transformed so that you get to the final (i.e. solution) vector.





For further info, you can check out the wikipedia article entitled 'Linear map' and scroll down to the subsection 'Matrices'.

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